Ergodic Properties of Continued Fraction Algorithms · Ergodic Properties of Continued Fraction...

119
Ergodic Properties of Continued Fraction Algorithms

Transcript of Ergodic Properties of Continued Fraction Algorithms · Ergodic Properties of Continued Fraction...

Ergodic Properties of Continued

Fraction Algorithms

Ergodic Properties of Continued Fraction Algorithms

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 3 februari 2003 om 13.30 uur

door

Yusuf HARTONO

Master of Science in Applied Mathematics,University of Missouri-Rolla, USA

geboren te Kundur, Indonesie.

Dit proefschrift is goedgekeurd door de promotor:Prof. dr. F. M. DekkingToegevoegd promotor: Dr. C. Kraaikamp

Samenstelling promotiecommisie:

Rector Magnificus voorzitterProf. dr. F. M. Dekking Technische Universiteit Delft, promotorDr. C. Kraaikamp Technische Universiteit Delft, toegevoegd promotorProf. dr. J. M. Aarts Technische Universiteit DelftProf. dr. M. Iosifescu Romanian Academy of Sciences, RoemenieProf. dr. F. Schweiger Universitat Salzburg, OostenrijkProf. dr. R. K. Sembiring Institut Teknologi Bandung, IndonesieDr. W. Bosma Katholieke Universiteit Nijmegen

The research in this thesis has beencarried out under the auspices ofthe Thomas Stieltjes Institute forMathematics, at the University ofTechnology in Delft.

Published and distributed by: DUP Science

DUP Science is an imprint ofDelft University PressP.O. Box 982600 MG DelftThe NetherlandsTelephone: +31 15 27 85 678Telefax: +31 15 27 85 706E-mail: [email protected]

ISBN 90-407-2381-8

Keywords: metric, arithmetic, continued fractions

Copyright c© 2002 by Y. HartonoAll rights reserved. No part of the material protected by this copyright noticemay be reproduced or utilized in any form or by any means, electronic or me-chanical, including photocopying, recording or by any information storage andretrieval system, without written permission from the publisher: Delft Univer-sity Press.

Cover designed by Silvia Yulianti

Printed in The Netherlands

To the memory of my mother

As far as the laws of mathematics refer to reality,they are not certain;

and as far as they are certain,they do not refer to reality.

Albert Einstein (1879 - 1955)

The beginning of knowledge isthe discovery of something we do not understand.

Frank Herbert (1920 - 1986), American Writer

The fear of the Lord is the beginning of knowledge, . . ..For from him and through him

and to him are all things.To him be the glory forever! Amen.

Proverbs 1:7; Romans 11:36

Acknowledgments

I am deeply indebted to many institutions and persons without whom this thesis wouldnever exist. At the first place I would like to express my gratitude and appreciation tomy supervisor Cor Kraaikamp, who introduced me to this exciting research area, forhis many interesting ideas and constant help that kept me in the right direction, forhis encouragement that kept me going, and for his patience that comforted me duringthe difficult time in the research. It has been a great pleasure to meet Karma andRafael. Also, I would like to express my gratefulness to my promotor Michel Dekkingfor inviting me to the Netherlands and providing me with the opportunity to carryout my doctoral research at Delft University of Technology (TU Delft). I should alsothank my committee for their valuable comments and suggestions.

The research leading to this thesis was a part of the scientific cooperation betweenDutch and Indonesian governments. I am very grateful to the program coordinatorsProf. Dr. R. K. Sembiring and Dr. A. H. P. van der Burgh who four and a halfyears ago organized a research workshop in Bandung that gave me a chance to beselected for this doctoral program at TU Delft. I thank Dr. O. Simbolon and Dr. B.Karyadi, former project managers of Proyek Pengembangan Guru Sekolah Menengah(PGSM) – (Secondary School Teachers Development Project), in Jakarta as well asDrs. P. Althuis, director of Center for International Cooperation in Applied Technology(CICAT) at TU Delft, for financial support and assistance during my research and stayin Holland.

I personally wish to thank all members of Afdeling CROSS, in particular vakgroupSSOR, for de gezelligheid and a very conducive atmosphere, especially Cindy, Ellen,and Diana for their both administrative and non-administrative assistance, and Carlfor his computer assistance. I also wish to thank Durk, Christel, Theda, Veronique,and Rene for a very wonderful friendship and for providing me with almost everythingI needed during my stay in Holland. My thanks should also go to all PGSM fellows(Abadi, Budi, Caswita, Darmawijoyo, Gede, Happy, Hartono, Kusnandi, Sahid, Siti,and Suyono) with whom I started the whole project together for everything we havehad together from fun to serious discussions about mathematics, particularly to Suyonowith whom I shared an office room and from whom I learned a lot about measuretheory; as well as to Agus, Komo and Julius with whom I spent some time sharing anapartment in Delft.

A lot of help and encouragement also came from my colleagues at the Facultyof Teacher Training and Education, particularly at the Department of MathematicsEducation, Sriwijaya University – Pak Ismail, Pak Purwoko, Bu Tri, and Somakim,just to name some, as well as from Kapten O. Tengke, Happy and other members ofBala Keselamatan Korps Kundur, and Kak Ibrahim. I would like to acknowledge themhere as well.

ix

x

I have appreciated all the suggestions, corrections, and comments during the prepa-ration of this thesis. I am the only one responsible for any mistake remaining in thisthesis.

I am also pleased to acknowledge all members of Het Leger des Heils Korps Delft,especially Majoor en mevrouw Loef and Kapitein en mevrouw Jansen, and of Interna-tional Student Chaplaincy, especially Fr. Ben and Rev. Stroh, for their generosity andwarm hospitality making me really feel at home and letting me participate in theirmany activities; to mention a few of them: Daniela, Bernardette, Carla, Riccardo,Yenory, Marco, Bibiana, Sandra, Sarah, Yadira, Susanne, Irek, Fabiana, Carmen,Ralph, Paul, Poni, Duleep, Nicolo, and Fabio. Many thanks to Mieke and Reini forconsidering me as kind aan huis in their home and family. Ik vind het heel leuk ommet Sara, David, en Nathan kennis te maken. Ik wil Tilak, Mart, en Jose, met allemedewerkers ook bedanken voor de samenwerking en een hele mooie vriendschap.

Moreover, I certainly enjoyed wonderful times together and very warm compan-ionship with Indonesian students, especially from TU Delft and IHE Delft, during mystudy − Bernadeta, Yusuf, Dwi, Teresia, Sri, Helena, Arief, Elisa, Raymon, Silvia,Dian, Dedy, Reiza, Evy, Joyce, Diah, Hilda, Firdian, Susi, Theresia, Zenlin, and Su-sana, just to mention a few. Their presence in my life shows that strong bonds oflove in service with each other are essential for developing a peaceful and harmoniouscommunity. They deserve an acknowledgement too.

I would like to dedicate this thesis to the memory of my mother who passed awayon January 3, 2001 at the age of 65. She taught me how to thank and have faith inGod, how to pray, and how to love and serve others. My father, who has taught methe meaning of hard-working life and, more importantly, how to do arithmetic duringmy early years at school, also deserves my special thanks for without him I could neverbe what I am now; and so does my aunt who took care of my son while my wife wasworking.

Saya ingin mempersembahkan tesis ini sebagai peringatan pada ibu saya yangmeninggal dunia pada tanggal 3 Januari 2001 pada usia 65. Dia telah mengajarsaya bagaimana bersyukur dan memiliki iman kepada Allah, bagaimana berdoa, danbagaimana mengasihi serta melayani orang lain. Ayah saya yang mengajar saya artikehidupan dan kerja keras dan, yang lebih penting lagi, mengajar saya berhitung padatahun-tahun awal saya di sekolah, juga pantas mendapat ucapan terima kasih khususkarena tanpa dia saya tidak mungkin menjadi seperti saya sekarang, demikian jugabibi saya yang mengurus anak saya saat istri saya bekerja.

Finally, a sincere expression of thanks should go to my dearest wife Elfarida andour beloved son Gabriel Ekoputra for their love, patience, and understanding duringmy absence in the family due to my study.

Terima kasih yang tulus saya sampaikan kepada istri saya dan anak kami tercintauntuk kasih, kesabaran, dan pengertian mereka selama saya tidak berada dalam kelu-arga.

Above all, I thank God for everything that has happened in my life.

Delft, October 17, 2002 Y.H.

Contents

1 Introduction 11.1 Some historical background and basic properties . . . . . . . . . 21.2 More recent developments . . . . . . . . . . . . . . . . . . . . . . 61.3 Some results in ergodic theory . . . . . . . . . . . . . . . . . . . . 81.4 Approximation coefficients . . . . . . . . . . . . . . . . . . . . . . 101.5 A brief description of the thesis . . . . . . . . . . . . . . . . . . . 12Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Odd Continued Fractions 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Insertions, singularizations and the OddCF . . . . . . . . . . . . 21

2.2.1 A singularization/insertion algorithm . . . . . . . . . . . . 212.2.2 Metrical properties of the OddCF . . . . . . . . . . . . . 232.2.3 Approximation coefficients . . . . . . . . . . . . . . . . . . 27

2.3 Grotesque continued fractions . . . . . . . . . . . . . . . . . . . . 322.4 Other odd continued fractions . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Maximal OddCF’s . . . . . . . . . . . . . . . . . . . . . . 342.4.2 Non-maximal expansions with odd digits . . . . . . . . . 35

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Engel Continued Fractions 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Ergodic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 On Ryde’s continued fraction with non decreasing digits . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Tong’s Spectrum for SRCF Expansions 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Tong’s pre-spectrum for the NICF . . . . . . . . . . . . . . . . . 644.3 Tong’s spectrum for Nakada’s α-expansions . . . . . . . . . . . . 69

4.3.1 The case g < α ≤ 1 . . . . . . . . . . . . . . . . . . . . . . 704.3.2 The case 1

2 ≤ α ≤ g . . . . . . . . . . . . . . . . . . . . . 734.4 Semi-regular continued fractions . . . . . . . . . . . . . . . . . . 77

xi

xii CONTENTS

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 A Note on Hurwitzian Numbers 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Hurwitzian numbers for the NICF . . . . . . . . . . . . . . . . . 825.3 Hurwitzian numbers for the backward continued fraction . . . . . 845.4 Hurwitzian numbers for α-expansions . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Mikowski’s DCF Expansions of Hurwitzian Numbers 936.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Minkowski’s DCF . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.3 DCF-Hurwitzian expansion . . . . . . . . . . . . . . . . . . . . . 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Samenvatting 101Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Curriculum Vitae 107

Chapter 1

Introduction

Mathematics is the queen of the science,and number theory is the queen of mathematics.

Carl Friedrich Gauss (1777 – 1855)

This thesis consists of five papers concerning continued fraction expansions inconnection with ergodic theory. Most of them deal with metrical properties ofcontinued fraction algorithms. Other aspects like approximation coefficients arealso studied here. Relationships among different continued fraction expansionsare developed via the singularization and insertion processes.

Throughout this thesis by a continued fraction expansion of any real numberwe mean an expression of the form

a0 +e1

a1 +e2

a2 +.. . +

en

an +.. .

, (1.1)

where a0 ∈ Z, an are positive integers and en ∈ R for n = 1, 2, . . .. In a muchmore convenient way we write (1.1) more compactly as

[ a0; e1/a1, e2/a2, · · · , en/an, · · · ] .The terms a1, a2, . . . are called the partial quotients of the continued fraction.The number of terms may be finite or infinite.

In case en = 1 for all n = 1, 2, . . ., we call (1.1) a regular continued fraction(RCF) expansion and write it as

[ a0; a1, a2, · · · , an, · · · ] . (1.2)

In general, let x be a real number, a0 = bxc, the largest integer not exceedingx, and write x = a0 + ξ. Now ξ = x− a0 ∈ [0, 1), and we write

x = [ a0; a1, a2, · · · ]

1

2 INTRODUCTION

if the RCF-expansion of ξ is given by

ξ = [ 0; a1, a2, · · · ].

In the next two sections some historical background and basic properties ofregular continued fractions will be presented, followed by a section on a fewbasic facts in ergodic theory and another on approximation coefficients. A briefdescription of the content of the papers in this thesis will conclude this chapter.

1.1 Some historical background and basic prop-erties

Continued fractions have a long history. It starts with the procedure knownas Euclid’s algorithm for finding the greatest common divisor (g.c.d.) of twointegers which occurs in the seventh book of Euclid’s Elements (c. 300 b.c.).This procedure is perhaps the earliest step towards the development of thetheory of continued fractions.

To see the relation between Euclid’s algorithm and (regular) continued frac-tions, consider Euclid’s algorithm for finding the g.c.d. of two integers a and bwith a > b > 0. We first let a0 = ba/bc. Putting

r1 := a− a0b, r0 := b ,

we have to find positive integers ai such that

ri−1 = airi + ri+1 , (1.3)

where 0 ≤ ri+1 < ri, for i = 1, 2, . . . until the procedure stops; that is, when wehave reached an index n such that rn 6= 0 and rn+1 = 0. In this case, we saythat rn is the g.c.d. of a and b.

Dividing (1.3) through by ri, we get

ri−1

ri= ai +

ri+1

ri, i = 1, 2, . . . n.

Writingri+1

ri=

1

ai+1 +ri+2

ri+1

, i = 1, 2, . . . , n,

and substituting it into the previous equation for each i yield

a

b= [ a0; a1, a2, · · · , an ],

which is the RCF expansion of a/b.

Some historical background and basic properties 3

To generalize Euclid’s algorithm to irrational numbers x in the unit interval,consider the continued fraction map T : [0, 1) → [0, 1) defined by

T (x) :=1x− b 1

xc, x 6= 0; T (0) := 0. (1.4)

Define further a1 = a1(x) = b1/xc and an = a1(Tn−1(x)), n = 0, 1, . . ., whereT 0(x) = x and Tn(x) = T (Tn−1(x)). We then have by (1.4)

x =1

a1 + T (x)= · · · = [ 0; a1, a2, · · · , an + Tn(x) ].

For rational numbers repeated application of T is in fact equivalent to Euclid’salgorithm. Hence, there exists an n0 ∈ N such that Tn0(x) = 0 and it followsthat a rational number has a finite RCF expansion. This is not the case forirrational numbers. If x is an irrational number, then Tn(x) is irrational for alln ≥ 0.

A finite truncation in (1.2) gives the so-called regular convergents

Pn

Qn= [ a0; a1, a2, · · · , an], n = 1, 2, . . . , (1.5)

where we assume that Qn > 0 and that gcd(Pn, Qn) = 1.The sequences (Pn)n≥−1 and (Qn)n≥−1 satisfy the following recursive for-

mulae

P−1 = 1, P0 = a0, Pn = anPn−1 + Pn−2 for n ≥ 1,

Q−1 = 0, Q0 = 1, Qn = anQn−1 + Qn−2 for n ≥ 1,

and the relationship

PnQn−1 − Pn−1Qn = (−1)n−1. (1.6)

Moreover, the regular convergents satisfy the following inequalities:

P0

Q0<

P2

Q2< · · · < P2n

Q2n< · · · < P2n+1

Q2n+1< · · · < P3

Q3<

P1

Q1.

For any irrational number x we say that (1.2) is the RCF expansion of x in case

limn→∞

Pn

Qn= x.

See, for instance, [O], [IK], and [HW] for more properties of RCF and proofs.

In general, (1.1) is called a semi-regular continued fraction (SRCF) in case a0 ∈Z, an are positive integers, and en = ±1 for all n ≥ 1, subject to the condition

en+1 + bn ≥ 1, for n ≥ 1,

4 INTRODUCTION

and with the restriction that in the infinite case

en+1 + bn ≥ 2, infinitely often.

Nakada’s α-expansions, for α ∈ [1/2, 1], are examples of SRCF expansions.Clearly, the RCF expansion (α = 1), the nearest integer continued fraction(NICF) expansion (α = 1/2), and Hurwitz’ singular continued fraction (g-expansion, with g = (

√5 − 1)/2,) are all SRCF expansions. Other examples

of SRCF expansions considered in this thesis are Minkowski’s diagonal contin-ued fraction (DCF) and odd continued fraction (OddCF).

In the area of applications, the great Dutch mathematician, mechanician, as-tronomer, and physicist, Christiaan Huygens (1629-1695) used the regular con-vergent to obtain the correct ratio for the rotations of planets when he designedthe toothed wheels of a planetarium. He described this in his Descriptio Auta-mati Planetarii, published posthumously in 1698. This is in fact a consequenceof the fact that continued fractions give the “best” rational approximations toirrational numbers.

The modern theory of continued fractions began with the writings of RafaelBombelli, born in about 1530 in Bologna. He showed, for example, in ourmodern notation, √

13 = [ 3; 4/6, 4/6, · · · ].Pietro Antonio Cataldi (1548-1626) also deserves some credits in continued frac-tions. He expressed √

18 = [ 4; 2/8, 2/8, · · · ].Using Euclid’s algorithm for finding the g.c.d. of 177 and 233, Daniel Schwenter(1585-1636) found the convergents 79/104, 19/25, 3/4, 1/1, and 0/1. It is prob-ably in Aritmetica Infinitorum (1655), a book by John Wallis, that the term con-tinued fraction was used for the first time. Great mathematicians such as Euler(1707-1783), Lambert (1728- 1777), Lagrange (1736-1813), Gauss (1777-1855),and many others also made important contributions to the earlier developmentof the theory of continued fractions. It is in particular Euler’s great memoir, DeFractionibus Continius (1737), that laid the foundation for the modern theory.See, for example, [O], [K1], [S], and [Di] for more history of continued fractions.

The metrical theory of continued fractions started with Gauss’ problem. Inhis diary on October 25, 1800, Gauss wrote (in modern notation) that

limn→∞

Fn(z) =log (z + 1)

log 2, z ∈ [0, 1), (1.7)

where Fn(z) = λ(Tn(x) < z), z ∈ [0, 1). Here T is the continued fraction mapdefined in (1.4) and λ denotes the Lebesgue measure. In a letter dated January30, 1812, he asked Laplace to prove (1.7) and to estimate the error-term

en(z) := Fn(z)− log (z + 1)log 2

.

Some historical background and basic properties 5

More than a century later this problem was solved by Kuzmin [Ku]. He showedin 1928 that

en(z) = O(q√

n) as n →∞for some constant q ∈ (0, 1). His proof is reproduced in Khinchine [Kh]. Oneyear later Levy independently proved that

|en(z)| < qn, n = 1, 2, . . .

with q = 3.5− 2√

2 = 0.67157 · · · . See Subsection 1.3.5 in [IK] for an improvedversion of Levy’s solution to Gauss’ problem. In 1961 P. Szusz used Kuzmin’sapproach to find that q = 0.485. Gauss’ problem was settled by Wirsing [Wi]who in 1974 found that q = 0.303 663 002 · · · . Results like these are now knownas Gauss-Kuzmin-Levy Theorems. The following result is a consequence of theseresults.

Theorem 1.1 (Levy, 1929) For almost all x ∈ [0, 1) with RCF expansion(1.2) one has

limn→∞

1n

log Qn =π2

12 log 2,

limn→∞

log(λ(∆n)) =−π2

6 log 2,

limn→∞

1n

∣∣∣∣x−Pn

Qn

∣∣∣∣ =−π2

6 log 2.

Here λ denotes the Lebesgue measure and ∆n = ∆n(i1, . . . , in) the so-calledfundamental interval defined by

∆n =

x ∈ [0, 1) :1

ij + 1≤ T j−1(x) <

1ij

, j = 1, 2, . . . , n

.

Moreover, among other things, Khintchine [Kh] showed the following.

Theorem 1.2 (Khintchine, 1935) For almost all x ∈ [0, 1) with RCF expan-sion (1.2) one has

limn→∞

(a1a2 · · · an)1/n =∞∏

k=1

(1 +

1k(k + 1)

) log klog 2

= 2.6854 · · · .

For proofs of the last two results see [DK]. One of them is proved in Section 1.3using some results in ergodic theory; see page 9.

The limiting distribution of Tn(x) in (1.7) leads us to a measure with density

1log 2

11 + x

, (1.8)

6 INTRODUCTION

today known as Gauss’ measure. This measure is invariant under the continuedfraction map T defined in (1.4) (i.e., T is Gauss measure preserving). To seethis, let (a, b) ⊂ [0, 1). Since

T−1(a, b) =(

1n + b

,1

n + a

),

we have, with γ denoting Gauss measure,

γ(T−1(a, b)) =1

log 2

∞∑n=1

∫ 1n+a

1n+b

dx

1 + x=

1log 2

logb + 1a + 1

= γ((a, b)).

See also Theorem 1.2.1 in [IK].

1.2 More recent developments

Another important development in the theory of continued fractions is the in-troduction of the so-called natural extensions by a group of Japanese mathe-maticians; see, e.g., the papers by H. Nakada, S. Ito and S. Tanaka [NIT], andNakada [N]. In this last paper the natural extension T of T is defined by

T (x, y) =(

T (x) ,1

a1(x) + y

), (x, y) ∈ [0, 1)× [0, 1]. (1.9)

It follows immediately that

Tn(x, y) = (Tn, Vn),

where Tn := Tn(x) and Vn := [0; an, an−1, · · · , a1] = Qn−1/Qn. Note that wemight consider Tn as the “future” of x at the “current” time n and Vn as the“past” of x up to time n. The points (Tn, Vn) are distributed in the unit squareaccording to the density function (log 2)−1(1 + xy)−2. In fact, this is a conse-quence of the ergodic system (1.11) in Theorem 1.6 on page 10.

Essential in this thesis are the so-called singularization and insertion pro-cesses by which we can obtain other SRCF expansions of x from its RCF expan-sion, such as the nearest integer continued fraction and odd continued fractionexpansions. We discuss some metrical properties of odd continued fraction ob-tained from the regular continued fraction via singularizations and insertions inChapter 2. The singularization process is based on the identity

A +e

1 +1

B + ξ

= A + e +−e

B + 1 + ξ,

while the insertion process rests on the identity

A +1

B + ξ= A + 1 +

−1

1 +1

B − 1 + ξ

,

More recent developments 7

where ξ ∈ [0, 1).This means, for example, that singularizing an+1 = 1 in an RCF expansion

(A) [ 0; a1, a2, · · · , an, 1, an+2, · · · ]with the sequence of convergents, say, (An/Bn)n≥1 results in an SRCF expansion

(B) [ 0; 1/a1, 1/a2, · · · , 1/(an + 1), −1/(an+2 + 1), · · · ].On the other hand, inserting −1/1 in the RCF (1.2) at (n + 1)-st position asan+2 6= 0 results in an SRCF expansion

(C) [ 0; 1/a1, 1/a2, · · · , 1/(an + 1), −1/1, 1/(an+2 − 1), · · · ].The effects of these two processes on the sequence of convergents were studiedin [K2]. It is shown that the sequence of convergents of the SRCF (B) can beobtained from that of the RCF (A) by removing An/Bn. On the other hand, thesequence of convergents of the SRCF (C) can be obtained from that of the RCF(1.2) by inserting (Pn + Pn−1)/(Qn + Qn−1) between Pn−1/Qn−1 and Pn/Qn.

In [K2] Kraaikamp introduced a new class of continued fractions called S-expansions which are obtained from the RCF only by using the singularizationprocess. The α-expansions (see [N]) are examples of S-expansions; see [IK] formore examples. Essential to these expansions is the so-called singularizationarea; that is, a subset S of [0, 1)× [0, 1] satisfying the following conditions.

(i) S ∈ B and S is a γ-continuity set,

(ii) S ⊆ [1/2, 1)× [0, 1],

(iii) S ∩ T (S) 6= ∅.To obtain the NICF of x, for instance, we have to singularize in each block

of m ∈ N∪ ∞ consecutive partial quotients equal to 1, the first, third,. . . etc.partial quotient. This leads to a singularization area

SNICF = [1/2, 1)× [0, g],

where g := (√

5 − 1)/2. Other two examples of S-expansions are Minkowski’sdiagonal continued fraction (DCF), with singularization area

SDCF =

(t, v) ∈ [0, 1)× [0, 1] :t

1 + tv>

12

,

and Bosma’s optimal continued fraction (OCF), with singularization area

SOCF =

(t, v) ∈ [0, 1)× [0, 1] : v < t and v <2t− 11− t

.

It was Wolfgang Doeblin [Do] who first discovered the ergodic system under-lying the RCF. Unfortunately, his results remained unnoticed for a long time.All classical results of continued fractions were obtained with probabilistic meth-ods until C. Ryll-Nardzewski showed in 1951 [R-N] how metrical results can beobtained in a more elegant way using ergodic theory. Some results in ergodictheory are presented in the next section.

8 INTRODUCTION

1.3 Some results in ergodic theory

Ergodic theory arose from an attempt in statistical mechanics to describe a sys-tem of a certain number of particles moving in a three-dimensional space at anygiven time. In general, let (Ω,B, P ) be a probability space. A transformationT : Ω → Ω is called measurable if T−1A ∈ B for all A ∈ B. We call T measurepreserving if it is measurable and P (T−1A) = P (A) for all A ∈ B. A transfor-mation T is said to be ergodic if every T -invariant subset of B has measure 0 or1, that is, T−1A = A ⇒ P (A) ∈ 0, 1. Equivalently, we say that (Ω,B, P, T )forms an ergodic system.

The following result is fundamental in ergodic theory; see, e.g., [P] and [Wa]for more details and proofs.

Theorem 1.3 (Birkhoff’s Individual Ergodic Theorem, 1931) Let (Ω,B, P )be a probability space and T : Ω → Ω a measure preserving transformation. Fur-ther, let f := Ω → R be such that f ∈ L1(Ω,B, P ). Then for almost all x

f∗(x) := limn→∞

1n

n−1∑

k=0

f(T kx)

exists. Moreover, we have f∗(x) ∈ L1(Ω,B, P ), f∗(x) = f∗(Tx), and∫Ω

f dP =∫Ω

f∗ dP .

The next theorem is an important consequence of Birkhoff’s ergodic theorem.

Theorem 1.4 Let (Ω,B, P, T ) be an ergodic system and f : Ω → R be such thatf ∈ L1(Ω,B, P ). Then for almost all x we have

limn→∞

1n

n−1∑

k=0

f(T k(x)) =∫

Ω

f dP.

The following fundamental result is very important in the development ofthe theory of continued fraction in connection with ergodic theory.

Theorem 1.5 Let Ω = [0, 1), B be the collection of all Borel sets of Ω, and γthe Gauss measure given in (1.8). Further, let T be the continued fraction map(1.4). Then

(Ω,B, γ, T ), (1.10)

forms an ergodic system.

The following example illustrates an application of Theorem 1.5.

Example 1.1 This equivalence can be easily checked for x ∈ [0, 1):

an(x) = a ⇔ Tn−1(x) ∈(

1a + 1

,1a

).

Some results in ergodic theory 9

Then the proportion of partial quotients equal to a in the sequence of partialquotients (an)n≥0 is for almost all x given by

1log 2

∫ 1a

1a+1

dx

1 + x=

1log 2

log(a + 1)2

a(a + 2).

This gives, for instance, that 2.272 · · · percent of the partial quotients equalto 7.

We now see that the results of Levy and Khintchine (see Theorems 1.1 and1.2) are corollaries of Theorem 1.5, together with Theorem 1.4. As an example,we give here a proof of Khintchine’s result; see also [DK].

Proof of Theorem 1.2. Define f(x) = log a1(x) where a1(x) = b1/xc, x ∈(0, 1). Then, due to ergodicity of T , we have

(a1a2 · · · an)1/n =

n−1∏

j=0

exp(f(T j(x)))

1/n

= exp

1

n

n−1∑

j=0

f(T j(x))

= exp

(∫ 1

0

f dλ

).

It remains to show that f is integrable. Now∫ 1

0

f dλ =∞∑

k=1

∫ 1k

1k+1

fdλ,

and∫ 1

k

1k+1

fdλ =1

log 2

∫ 1k

1k+1

log a1(x)1 + x

dx =log k

log 2log

(1 +

1k(k + 2)

)∼ log k

k(k + 2)

as k →∞. Here we have used limε→0(1+ ε)/2ε = 1. The result follows from thefact that ∞∑

k=1

log k

k(k + 2)

is convergent and writing

∞∑

k=1

log k

log 2log

(1 +

1k(k + 2)

)= log

∞∏

k=1

(1 +

1k(k + 2)

) log klog 2

.

2

The natural extension of the system (1.10), which is used several times inthis thesis, is given in the next theorem. More details on this result can befound in [NIT] and [N].

10 INTRODUCTION

Theorem 1.6 Let Ω = Ω × [0, 1], B be the class of all Borel sets of Ω, γ bethe extended (two-dimensional) Gauss measure defined by

γ(A) =1

log 2

A

dx dy

(1 + xy)2, A ∈ Ω,

and T is the natural extension (1.9) of T . Then

(Ω, B, γ, T ), (1.11)

forms an ergodic system.

1.4 Approximation coefficients

One of the most important reasons to use (regular) continued fractions is thatcontinued fractions yield “the best” rational convergents to irrational numbers.In order to express the quality of approximation of an irrational number x by arational number p/q, we introduce the approximation coefficient θ(x, p/q) by

θ(x, p/q) = q|qx− p|.

A classical theorem by Borel now states that for every irrational x there areinfinitely many rationals p/q such that θ(x, p/q) < 1/

√5.

For any irrational number x we define the approximation coefficients θn by

θn := θn(x) = Qn|Qnx− Pn|, n = 1, 2, . . . . (1.12)

They measure how well the rational number Pn/Qn approximates an irrationalnumber x. Since it can be shown that

∣∣∣∣x−Pn

Qn

∣∣∣∣ <1

Q2n

,

we immediately see that 0 < θn < 1 for all n ≥ 1. Using

x =Pn + Pn−1Tn

Qn + Qn−1Tn,

in (1.12), we can show that

θn =Tn

1 + TnVn, and θn−1 =

Vn

1 + TnVn. (1.13)

Hence, defining φ : Ω → R2 by

φ(x, y) =(

y

1 + xy,

x

1 + xy

)

Approximation coefficients 11

leads to the fact that(θn−1, θn) = φ(Tn, Vn). (1.14)

In fact, φ(Ω) = ∆, where ∆ is a triangle with vertices (0,0), (1,0), and (0,1). Itthen follows immediately that

θn−1 + θn < 1, n = 1, 2, . . . ,

and hence

min(θn−1, θn) <12, n = 1, 2, . . . ,

which is a well-known result due to Vahlen [V].Using the fact that

(θn, θn+1) = φ(T (φ−1(θn−1, θn))),

Jager and Kraaikamp [JK] were able to show that

θn+1 = θn−1 + an+1

√1− 4θn−1θn − a2

n+1θn.

From this it easily follows that

min(θn−1, θn, θn+1) <1√

a2n+1 + 4

and

max(θn−1, θn, θn+1) >1√

a2n+1 + 4

.

The former clearly generalizes Borel’s classical result, the latter was found byJ.C. Tong [T1]. As a corollary we find the following result.

Theorem 1.7 For all irrational numbers x and all n ≥ 0 one has

min(θn−1, θn, θn+1) <1√5;

the constant 1/√

5 cannot be replaced by a smaller one.

The following result is another consequence of (1.14) together with Theo-rem 1.6.

Theorem 1.8 (Jager, 1986) The sequence (θn−1, θn) are distributed over thetriangle ∆ according, for almost all x, to the density function

f(a, b) =1

log 21√

1− 4ab.

12 INTRODUCTION

See [J] for details.

Continued fractions play an important role in the theory of prime-testing (see,e.g., Bressoud’s book [B]). In 1981, H. W. Lenstra conjectured that for almostall x

limn→∞

1n

#j : 1 ≤ j ≤ n, θj(x) ≤ z, where 0 ≤ z ≤ 1, (1.15)

exists and equals F (z), where

F (z) =

zlog 2 , 0 ≤ z ≤ 1

2 ,

1log 2(1− z + log 2z), 1

2 ≤ z ≤ 1.

In fact (1.15) had been conjectured in 1940 by Wolfgang Doeblin [Do]. In 1984Knuth [Kn] showed that

limN→∞

1N

#1 ≤ i ≤ N : θ ≤ θi ≤ θ + d θ =1

log 2

∫ θ+d θ

θ

`(t) dt,

where

`(t) = min(

1,1t− 1

).

In 1983 Bosma, Jager, and Wiedijk [BJW] proved the Lenstra-Doeblin conjec-ture using Nakada’s natural extension (Ω, B, γ, T ).

1.5 A brief description of the thesis

This thesis consists of five papers dealing with continued fractions.

Chapter 21 is concerned with the continued fraction with odd partial quotients(OddCF). The relation between OddCF and RCF is developed via singulariza-tion and insertion processes. Using Schweiger’s natural extension for the OddCFwe show that the sequence of convergents of the nearest integer continued frac-tion (NICF) is a subsequence of that of OddCF. Using the method in [JK],we obtain a result for OddCF approximation coefficients which coincides withTong’s result for NICF [T2]. Through the relation between RCF and grotesquecontinued fraction (GCF) developed again via singularizations and insertionswe see that the sequence of GCF convergents forms a subsequence of that ofHurwitz’ singular continued fraction. Maximal and non-maximal OddCF arealso discussed.

1a joint work with Cor Kraaikamp in Rev. Romaine Math. Pures Appl. 47 (2002), no. 1.

A brief description of the thesis 13

In Chapter 32 we consider the map TE : [0, 1) → [0, 1) given by

TE(x) :=1b 1

xc

(1x− b 1

xc)

, x 6= 0; TE(0) := 0.

This map yields a (unique) continued fraction expansion of x ∈ [0, 1) with non-decreasing partial quotients of the form

1

b1 +b1

b2 +b2

b3 +.. . +

bn−1

bn +.. .

, bn ∈ N, with bn ≤ bn+1 .

We call this expansion Engel continued fraction (ECF) expansion of x since themap TE is a modified version of the Engel series expansion map.

Some basic properties of RCF also hold for the ECF but they differ in manyways. For instance, ECF convergents behave differently from regular ones. Itturns out that TE is ergodic with respect to Lebesgue measure but has nofinite invariant measure, equivalent to Lebesgue. Moreover, it is shown that themap TE has infinitely many σ-finite, infinite invariant measures, two of whichare given here. Additionally, we relate the ECF to Ryde’s monotonen, nicht-abnehmenden Kettenbruch (MNK) generated by the map TR : ( 1

2 , 1) → (12 , 1),

given by

TR(x) = SR(x) =k

x− k, for x ∈ R(k) :=

(k

k + 1,k + 1k + 2

), k ∈ N,

through an isomorphism. From this it follows, for example, that the map TR

is ergodic with respect to Lebesgue measure but no finite TR-invariant measureequivalent to Lebesgue exists and that not every quadratic irrational has anultimately periodic ECF expansion.

A Hurwitz-type spectrum was studied for the nearest integer continued fractionby Jager and Kraaikamp in [JK]. With (Θn)n≥1 denoting the sequence of NICFapproximation coefficients they showed that

min(Θn−1, Θn,Θn+1) <52(5√

5− 11) = 0.4508 · · · .

In [T2] Tong extended this result and proved that

min(Θn−1, Θn, . . . , Θn+k) <1√5

+1√5

(3−√5

2

)2k+3

.

2a joint work with Cor Kraaikamp and F. Schweiger in J. de Theorie des Nombres deBordeaux 14 (2002).

14 INTRODUCTION

Chapter 43 gives a proof of Tong’s result using the method from [JK] whichyields some metrical observations with respect to Tong’s spectrum. General-izations to a larger class of semi-regular continued fraction expansions are alsoderived.

A number x ∈ R is called Hurwitzian if its RCF expansion (1.2) can be writtenas

x = [a0; a1, · · · , an, an+1(k), · · · , an+p(k)]∞k=0,

where an+1(k), . . . , an+p(k) ( the so-called quasi period of x) are polynomialswith rational coefficients which take positive integral values for k = 0, 1, 2, . . .,and at least one of them is not constant. This clearly generalizes periodic con-tinued fractions. In Chapter 54 we define the Hurwitzian numbers for the NICF,the ‘backward’ continued fraction expansion, and α-expansions. We show thatthe set of Hurwitzian numbers for such continued fractions coincides with theclassical set of Hurwitzian numbers.

Chapter 65 is a continuation of the previous chapter. In this chapter we defineHurwitzian numbers for Minskowski’s diagonal continued fraction (DCF). Wealso show that the set of DCF-Hurwitzian numbers coincides the classical setof Hurwitzian numbers. The situation is more complicated here than in theprevious paper due to the difference in shape of the singularization area of theNICF (and other α-expansions) on one hand, and that of the DCF on the other.

3a joint work with Cor Kraaikamp4a joint work with Cor Kraaikamp in Tokyo J. Math. 25 (2002), no. 25J. Matematika atau Pembelajarannya VIII (2002), 837–841.

Bibliography

[B] Bressoud, D. M. – Factorization and Primality Testing, Sringer-Verlag, NewYork, (1989). MR 91e:11150

[BJW] Bosma, W., H. Jager and F. Wiedijk. - Some metrical observations onthe expansion by continued fractions, Indag. Math. 45 (1983), 353–379.

[DK] Dajani, K. and C. Kraaikamp. – Ergodic Theory of Numbers, Carus Math-ematical Monographs, No. 29, (2002).

[Di] Dickson, L. E. – History of the Theory of Numbers, Vols I, II, III, CarnegieInstitution of Washington, Washington, (1991-1923).

[Do] Doeblin, W. — Remarque sur la theorie metrique des fractions continues,Compositio Math. 7 (1940), 353–371. MR 2,107e

[HW] Hardy, G. H. and Wright, E. M. – An introduction to the theory ofnumbers. Fifth edition. The Clarendon Press, Oxford University Press, NewYork, (1979). MR 81i:10002

[IK] Iosifescu, M. and C. Kraaikamp. - The Metrical Theory of Continued Frac-tions, Kluwer Academic Press, Dordrecht, The Netherlands, (2002).

[J] Jager, H. - Continued fraction and ergodic theory, Transcendental Numbersand Related Topics, RIMS Kokyuroku, 599, Kyoto University, Kyoto, Japan,(1986), 55–59.

[JK] Jager, H. and C. Kraaikamp. — On the approximation by continued frac-tions, Indag. Math. 51 (1989), no. 2, 289-307. MR 90k:11084

[Kh] Khintchine, A. Ya. - Metrische kettenbruchprablemen , Compositio Math.1 (1935), 361–382.

[Kn] Knuth, D. E. - The distribution of continued fraction approximations, J.Number Theory 19 (1984), no 3, 443–448. MR 86d:11058

[K1] Kraaikamp, C. - Metric and Arithmetic Results for Continued FractionExpansions, Ph.D. Thesis (1990), Universiteit van Amsterdam, Amsterdam.

15

16 INTRODUCTION

[K2] Kraaikamp, C. - A new class of continued fraction expansions, Acta Arith.57 (1991), no. 1, 1–39. MR 92a:11090

[Ku] Kuzmin, R. O. - On a problem of Gauss , Dokl. Akad. Nauk. SSSR Ser.A, (1928), 375–380.

[N] Nakada, H. — Metrical theory for a class of continued fraction transforma-tions and their natural extensions, Tokyo J. Math. 4 (1981), no. 2, 399–426.MR 83k:10095

[NIT] Nakada, H., S. Ito and S. Tanaka — On the invariant measure for thetransformations associated with some real continued fractions, Keio EngrgRep., 30 (1977), no. 13, 159–175. MR 58 16574

[O] Olds, C. D. – Continued fractions, Random House, New York, (1963). MR26#3672

[P] Petersen, K. – Ergodic Theory, Cambridge University Press, Cambridge,(1997).

[R-N] Ryll-Nardzewski, C. - On the ergodic theorems. II. Ergodic theory of con-tinued fractions, Studia Math. 12 (1951), 74–79.

[S] Schweiger, F. – Ergodic theory of fibred systems and metric number theory,Oxford Science Publications. The Clarendon Press, Oxford University Press,New York, (1995). MR 97h:11083

[T1] Tong, Jing Cheng – The conjugate property of the Borel theorem on Dio-phantine approximation, Math. Z. 184 (1983), no. 2, 151–153. MR 85m:11039

[T2] Tong, Jing Cheng — Approximation by nearest integer continued fractions(II), Math. Scand. 74 (1994), no. 1, 17–18. MR 95c:11085

[V] Vahlen, K. Th. — Uber Naherungswerte und Kettenbruche, Journal f. d.reine und angew. Math. 115 (1895), 221–233.

[Wa] Walters, P. – An Introduction to Ergodic Theory, Springer-Verlag NewYork, Inc., New York, (2000).

[Wi] Wirsing, E. – On the theorem of Gauss-Kuzmin-Lery and a Frobenius typetheorem for function spaces, Acta Arith. 24 (1974), 507–528. MR 49 2637

Chapter 2

Odd Continued Fractions

2.1 Introduction

It is well-known that every x ∈ [0, 1) can be written as a finite (in case x isrational) or infinite (when x is irrational) continued fraction with odd partialquotients:

x =e1

a1 +e2

a2 +.. . +

en

an +.. .

=: [ 0; e1/a1, e2/a2, · · · , en/an, · · · ] , (2.1)

where e1 = 1, ei = ±1 and ai is a positive odd integer, for i ≥ 1, and

ai + ei+1 > 1, i ≥ 1.

We call (2.1) the odd continued fraction (OddCF) expansion of x. Apart fromthe OddCF-expansion of x one also has the so-called grotesque continued fraction(or GCF) expansion of any x ∈ [G − 2, G), where G is the golden mean, i.e.,G = 1

2 (√

5 + 1). The GCF-expansion is also given by (2.1), again with oddpartial quotients ai and ei = ±1, but now these ai and ei must satisfy

ai + ei > 1, i ≥ 1,

and e1 = sgn(x).There is an extended literature on both the OddCF and the GCF. In two

(unpublished) papers F. Schweiger obtained the ergodic theorem underlying theOddCF and its natural extension − where, as a by-result − he showed that theGCF is the dual algorithm of the OddCF ([S1], [S2]), and studied the approxi-mation properties of the OddCF ([S2]). Around the same time G.J. Rieger alsoobtained a Gauss-Kuzmin theorem for the OddCF and found the ergodic sys-tems underlying both the OddCF and the GCF ([R2]). Also a Heilbron-theorem

17

18 ODD CONTINUED FRACTIONS

was given by Rieger for both expansions in [R1]. In two recent papers G.I. Sebereturned to the convergence rate in the Gauss-Kuzmin problem for the OddCF([Se1]) and GCF ([Se2]) using the theory of random systems with complete con-nections. Sebe also obtained the natural extension for the GCF. More resultson the OddCF and the GCF can be found in papers by S. Kalpazidou ([Ka1],[Ka2]) and D. Barbolosi ([B1], [B2]).

At first sight one might be tempted to say that nothing can be said anymoreabout these expansions! In [B2], Barbolosi showed that for any x ∈ [0, 1) thesequence of nearest integer continued fraction (NICF) convergents of x formsa subsequence of the sequence of OddCF-convergents of x. In order to un-derstand this result we were led to a new class of continued fraction expansionswith odd partial quotients, of which the OddCF and the GCF are two examples.

In general, a semi-regular continued fraction (SRCF) is a finite or infinitefraction

b0 +e1

b1 +e2

b2 +.. . +

en

bn +.. .

= [ b0; e1/b1, e2/b2, · · · , en/bn, · · · ] , (2.2)

with en = ±1; b0 ∈ Z; bn ∈ N, for n ≥ 1, subject to the condition

en+1 + bn ≥ 1, for n ≥ 1, (2.3)

and with the restriction that in the infinite case

en+1 + bn ≥ 2, infinitely often. (2.4)

A finite truncation in (2.2) yields the SRCF-convergents

An/Bn := [ b0; e1/b1, e2/b2, · · · , en/bn] ,

where it is always assumed that gcd(An, Bn) = 1. We say that (2.2) is anSRCF-expansion of an irrational number x in case

x = limn→∞

An

Bn.

Clearly the OddCF is an example of an SRCF-expansion, but the GCF is not.Other examples of SRCF-expansions are the nearest integer continued fraction(NICF) expansion, satisfying

en+1 + bn ≥ 2 for n ≥ 1,

and Hurwitz’ singular continued fraction (HSCF) expansion, which satisfies

en + bn ≥ 2 for n ≥ 1.

Introduction 19

Perhaps the best-known example of an SRCF-expansion is the so-called regularcontinued fraction expansion (RCF); every real irrational number x has a uniqueRCF-expansion

d0 +1

d1 + 1

d2 +.. .

=: [ d0; d1, d2, · · · ], (2.5)

where d0 ∈ Z is such that x− d0 ∈ [0, 1), and dn ∈ N for n ∈ N.

Obviously the GCF is not an SRCF, but a so-called unitary expansion, seealso [G]. Unitary expansions are defined in a way similar to SRCF-expansions,the difference being that (2.3) and (2.4) are replaced by

en + bn ≥ 1, for n ≥ 1,

and with the restriction that in the infinite case

en + bn ≥ 2, infinitely often.

Essential in our investigations are the notions of insertion and singularizationof a partial quotient equal to 1, which were studied in detail in [K].

A singularization is based upon the identity

A +e

1 +1

B + ξ

= A + e +−e

B + 1 + ξ,

where ξ ∈ [0, 1).

To see the effect of a singularization, let (2.2) be an SRCF-expansion of x. Afinite truncation yields the sequence of convergents (rk/sk)k≥−1. Suppose thatfor some n ≥ 0 one has

bn+1 = 1; en+2 = 1 ,

i.e.,

[ b0; e1/b1, · · · ] = [ b0; e1/b1, · · · , en/bn, en+1/1, 1/bn+2, · · · ]. (2.6)

The transformation σn which changes this continued fraction (2.6) into thecontinued fraction

[ b0; e1/b1, · · · , en/(bn + en+1),−en+1/(bn+2 + 1), · · · ], (2.7)

which is again a continued fraction expansion of x, with convergents, say(pk/qk)k≥−1, is called a singularization. It was shown in [K] that the sequence

of vectors(

pk

qk

)

k≥−1

is obtained from(

rk

sk

)

k≥−1

by removing the term(

rn

sn

)

from the latter.

20 ODD CONTINUED FRACTIONS

An operation which is in some sense the ‘opposite’ of a singularization is aso-called insertion. An insertion is either based upon the identity

A +1

B + ξ= A + 1 +

−1

1 +1

B − 1 + ξ

,

or on the identity

A +1

B + ξ= A− 1 +

1

1 +− 1

B + 1 + ξ

,

where ξ ∈ [0, 1).Let (2.2) be an SRCF-expansion of x, and suppose that for some n ≥ 0 one

hasbn+1 > 1; en+1 = 1.

An srcf-insertion is the transformation τn which changes (2.2) into

[ b0; e1/b1, · · · , en/(bn + 1), −1/1, 1/(bn+1 − 1), · · · ],

which is again an SRCF-expansion of x, with convergents, say, (pk/qk)k≥−1. Let(rk/sk)k≥−1 be the sequence of convergents connected with (2.2). Using some

matrix-identities it was shown in [K] that the sequence of vectors(

pk

qk

)

k≥−1

is

obtained from(

rk

sk

)

k≥−1

by inserting the term(

rn + rn−1

sn + sn−1

)before the n-th

term of the latter sequence, i.e.,(

pk

qk

)

k≥−1

≡(

r−1

s−1

),

(r0

s0

), . . . ,

(rn−1

sn−1

),

(rn + rn−1

sn + sn−1

),

(rn

sn

),

(rn+1

sn+1

), . . . .

An srcf-insertion is denoted by −1/1.

Now let (2.2) be a unitary-expansion of x with the sequence of convergents(rk/sk)k≥−1. Suppose that for some n ≥ 0 one has

bn > 1; en+1 = 1.

Applying the second insertion-identity changes (2.2) into

[ b0; e1/b1, · · · , en/(bn − 1), 1/1, −1/(bn+1 + 1), · · · ],

which is again a unitary-expansion of x, with convergents, say, (pk/qk)k≥−1.This kind of insertion is called a unitary-insertion. In this case the sequence

Insertions, singularizations and the OddCF 21

of vectors(

pk

qk

)

k≥−1

of the new expansion is obtained from(

rk

sk

)

k≥−1

by in-

serting the term(

rn − rn−1

sn − sn−1

)before the n-th term of the latter sequence. A

unitary-insertion is denoted by 1/1−.

By combining the operations of singularization and srcf/unitary-insertion onecan obtain any semi-regular/unitary continued fraction expansion of a numberx from its RCF expansion. In [K] a whole class of semi-regular continued frac-tions was introduced via singularizations only (some of these SRCF’s were new,some classical − like the continued fraction to the nearest integer, or Hurwitz’singular continued fraction (HSCF)), and their ergodic theory studied (the mainidea in [K] is that the operation of singularization is equivalent to having aninduced map on the natural extension of the RCF).

In the next section we will show that the OddCF-expansion can be obtainedfrom the RCF via suitable srcf-insertions and singularizations. We also willderive some metrical results for the approximation coefficients of the OddCF.In Section 2.3 we will see that the GCF can be obtained from the RCF viasingularizations and unitary-insertions. This will lead us in Section 2.4 to a newclass of semi-regular/unitary continued fraction expansions with odd partialquotients.

2.2 Insertions, singularizations and the OddCF

2.2.1 A singularization/insertion algorithm

The following theorem describes an algorithm which turns the RCF-expansionof any x ∈ [0, 1) into the OddCF-expansion of x. The proof of this theoremfollows easily by inspection, and is therefore omitted.

Theorem 2.1 Let x ∈ [0, 1) with RCF-expansion (2.5), i.e., d0 = 0. Thenstarting from the RCF-expansion (2.5) of x, the following algorithm yields theOddCF-expansion of x.

(I) Let m := infn ∈ N; dn is even .(i) If dm+1 > 1, insert −1/1 after dm to obtain

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm + 1), −1/1, 1/(dm+1 − 1), 1/dm+2, · · · ] .

(ii) If dm+1 = 1, let k := infn > m; dn > 1 (k = ∞ is allowed). Nowsingularize in the block of partial quotients

dm+1 = 1, dm+2 = 1, . . . , dk−1 = 1

22 ODD CONTINUED FRACTIONS

the first, third, fifth, etc. partial quotients equal to 1, to arrive at

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm + 1), −1/3, · · · , −1/3| z k−m−2

2 −times

, −1/(dk + 1), 1/dk+1, · · · ] ,

in case k −m− 1 is odd or k = ∞; in the latter case we find

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm + 1), −1/3, · · · , −1/3, · · · ] .In case k −m− 1 is even we obtain

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm + 1), −1/3, · · · , −1/3| z k−m−3

2 −times

, −1/2, 1/dk, 1/dk+1, · · · ] .

In this case insert −1/1 to arrive at

[ 0; 1/d1, · · · , 1/(dm + 1), −1/3, · · · , −1/3| z k−m−1

2 −times

, −1/1, 1/(dk − 1), 1/dk+1, · · · ] ,

(II) Let m ≥ 1 be the first index in the new SRCF-expansion [ c0; e1/c1, · · · ] ofx obtained in (I) for which cm is even. Repeat the procedure from (I) tothis new SRCF-expansion of x with this value of m.

As soon as m = ∞ in (II) we have obtained the OddCF-expansion of x.

The following example illustrate how to use Theorem 2.1

Example 2.1 Let x ∈ [0, 1) have RCF-expansion

[0; 1/3, 1/1, 1/4, 1/7, 1/1, 1/1, 1/1, 1/1, 1/1, 1/1, 1/1, 1/1, 1/5, · · · ].

(i) Apply the algorithm with m = 3. Since d4 > 1, we insert −1/1 after 1/4to obtain

[0; 1/3, 1/1, 1/5,−1/1, 1/6, 1/1, 1/1, 1/1, 1/1, 1/1, 1/1, 1/1, 1/1, 1/5, · · · ].

(ii) Apply the algorithm with m = 5 in the new expansion. Since d6 = · · · =d11 = 1, we singularize d6, d8, d10 and d11 to arrive at

[0; 1/3, 1/1, 1/5,−1/1, 1/7,−1/3,−1/3,−1/3,−1/2, 1/5, · · · ].

Now insert −1/1 to arrive at

[0; 1/3, 1/1, 1/5,−1/1, 1/7,−1/3,−1/3,−1/3,−1/3,−1/1, 1/4, · · · ].

(iii) Apply the algorithm with m = 10 in this new expansion and continueuntil m = ∞.

Insertions, singularizations and the OddCF 23

2.2.2 Metrical properties of the OddCF

In [S2] (and implicitly in [S1]), Schweiger introduced and studied the naturalextension of the ergodic system underlying the OddCF. See also [Se2], whereSebe obtained the natural extension of the GCF.

SettingB(+, k) =

(12k

, 12k − 1

], k = 1, 2, . . . ,

B(−, k) =(

12k − 1 , 1

2k − 2

], k = 2, 3, . . . ,

the map

T (x) = e ·(

1x− (2k − 1)

), x ∈ B(e, k), e = ±1,

generates the OddCF-expansion (2.1) of x. Notice that e1 = 1 and that en =en(x), an = an(x) are given by

(en+1, an) = (e, 2k − 1) ⇔ Tn−1(x) ∈ B(e, k), for n ≥ 1.

The dual-algorithm T ∗ of T is given by

T ∗(x) =ε(x)x

− (2k − 1), ε(x) = sgn(x),

on an appropriate partition of [G − 2, G]. This dual-algorithm is the map un-derlying the GCF. Setting

Ω = [0, 1)× [G− 2, G]

and defining T : Ω → Ω by

T (x, y) =(

T (x),e

a + y

),

where e = ±1 and a = 2k − 1 are such that x ∈ B(e, k), Schweiger showed that

(Ω,B, µ, T )

forms an ergodic system. Here B is the collection of Borel sets of Ω, and µ isa probability measure on (Ω,B), with density (3 log G)−1(1 + xy)−2 on Ω, seealso Figure 2.1.

In [K] it was shown that the nearest integer continued fraction (NICF) expansionof any x ∈ [0, 1) can be obtained from the RCF-expansion of x by applying thefirst step in (I)(ii) of Theorem 2.1 to any block of regular partial quotients equalto 1, which is preceded and followed by a regular partial quotient different from1 (this restriction does not apply if the expansion of x starts with 1’s, or whenthe block of 1’s is infinite). But then Barbolosi’s result from [B2], which statesthat the sequence of NICF-convergents of x forms a subsequence of the sequenceof OddCF-convergents of x is an immediate corollary of Theorem 2.1.

We have the following theorem.

24 ODD CONTINUED FRACTIONS

−g2

12

10

g

1

G

Legends:

= NICF-singularization area

= insertion area

Figure 2.1: Schweiger’s Natural Extension (Ω,B, µ, T )

Theorem 2.2 Let x ∈ [0, 1) be an irrational number with OddCF-expansion(2.1), with RCF-expansion (2.5), and let

x = [ b0; f1/b1, · · · ]be the NICF-expansion of x. Say

(pn/qn)n≥−1, (Pn/Qn)n≥−1, and (An/Bn)n≥−1

are the sequences of the OddCF, RCF resp. the NICF-convergents of x.

(i) Then there exists an arithmetical function k = k(x) : N→ N, such that

An

Bn=

pk(n)

qk(n), n ≥ 1,

and one has for almost all1 x that

limn→∞

k(n)n

=32

.

(ii) In the OddCF-expansion (2.1) of x, singularize every digit ai = 1 forwhich ei = −1 (i.e., remove all the inserted mediant convergents). Indoing so we find an SRCF-expansion

x = [ 0; ε1/u1, ε2/u2, · · · ]1All almost all statements are with respect to the Lebesgue measure λ.

Insertions, singularizations and the OddCF 25

of x, with convergents, say, Cn/Dn, n ≥ −1. Then (Cn/Dn)n≥−1 formsa subsequence of (Pn/Qn)n≥−1 and has (An/Bn)n≥−1 as a subsequence.There exist arithmetical functions `, `∗ : N→ N such that

Cn

Dn=

P`(n)

Q`(n),

An

Bn=

C`∗(n)

D`∗(n), n ≥ 1,

and one has for almost all x that

limn→∞

`(n)n

=log 4

log 2G, lim

n→∞`∗(n)

n=

log 2G

2 log G.

Remark 2.1 Let x ∈ [0, 1) be an irrational number with RCF-convergents(Pn/Qn)n≥−1 and NICF-convergents (An/Bn)n≥−1. Since (An/Bn)n≥−1 formsa subsequence of (Pn/Qn)n≥−1, there exists an arithmetical function w = w(x) :N→ N such that

An

Bn=

Pw(n)

Qw(n), n ≥ 1.

In [A], W.W. Adams showed that

limn→∞

w(n)n

=log 2log G

= 1.4404 · · · a.e.,

see also [J] and [K]. In words, this result states that for almost all x about30.58% of the regular convergents of x were removed (in an appropriate way)from the RCF-expansion of x to obtain the NICF-expansion of x.

Proof.(i) Let Snicf and Sins be the NICF-singularization and insertion areas, respec-tively; see Figure 2.1. Let

S = Snicf ∪ Sins.

Then a simple calculation yields that

µ(S) =13.

The result follows from taking S as the singularization area and using Theo-rem (4.13) in [K].(ii) Consider ∆ = Ω \ Sins and S : ∆ −→ ∆ defined by

S(x, y) =

T (x, y), T (x, y) /∈ Sins;

T 2(x, y), T (x, y) ∈ Sins.

It follows from ergodicity of T that

(∆,B, µ∆,S),

26 ODD CONTINUED FRACTIONS

whereµ∆(E) =

1(1− µ(Sins))3 log G

E

dx dy

(1 + xy)2,

forms an ergodic system. Taking Snicf as the singularization area in this system,together with

µ∆(Snicf) =log 2g

log 2G= 0.18046 · · · ,

one obtains, see Theorem (4.13) in [K],

limn−→∞

`∗(n)n

=1

1− µ∆(Snicf)=

log 2G

2 log G= 1.2202 · · · .

By construction (Cn/Dn)n≥−1 forms a subsequence of the sequence of RCF-convergents. Thus there exist arithmetical functions `, `∗ : N −→ N such that

Cm

Dm=

P`(m)

Q`(m),

Am

Bm=

C`∗(m)

D`∗(m), for m ∈ N.

From Remark 2.1 we have

w(n)n

=`(`∗(n))`∗(n)

`∗(n)n

,

which gives

limn−→∞

`(`∗(n))`∗(n)

=log 4

log 2G= 1.1804 · · ·

for almost all x. It now remains to show that `(m)/m converges to the same limitlog 4/ log 2G as m −→∞, where m ∈ N is from the set of indices of convergents(Cn/Dn) which were removed to obtain (An/Bn)n≥−1. To this end, let m ∈ Nbe such that for all n ∈ N

m 6= `∗(n).

But then there exists an n ∈ N such that

m− 1 = `∗(n) and m + 1 = `∗(n + 1) (2.8)

since otherwise two consecutive convergents in (Cn/Dn)n≥−1 (corresponding toinsertions) would have been singularized (which is impossible, since we neverhave two consecutive insertions). Due to the fact that two consecutive NICF-convergents cannot be too far apart in the sequence of RCF-convergents, wemust have that

`(m− 1) = `(m)− 1 and `(m + 1) = `(m) + 1.

Hence, we have

`(m− 1)m− 1

m− 1m

=`(m)− 1

m<

`(m)m

<`(m) + 1

m=

`(m + 1)m + 1

m + 1m

and the result follows from (2.8). 2

Insertions, singularizations and the OddCF 27

Remark 2.2 1. Let S be defined as in the proof of Theorem 2.2 (i). Then dueto Abramov’s formula (see [P], pp. 257-258) one has

h(S) =h(T )µ(S)

,

It is well-known, see e.g. [N], that the entropy h(Tnicf) of the NICF equalsπ2/6 log G, and since (Ω \ S, 1

µ(S)µ,S) forms an ergodic system which is metri-cally isomorphic to the ergodic system underlying the NICF, we find that

h(S) =π2

6 log G= 3.418 · · · ,

2. In Theorem 2.2(ii) we saw that about 15.29% of the RCF-convergents wereremoved in the algorithm from Theorem 2.1 to get the OddCF-expansion ofx, and that 18.64% of the OddCF-convergents were obtained by appropriateinsertions.

Taking Sins as the singularization area, we immediately see that(Cn/Dn)n≥−1 forms a subsequence of (pn/qn)n≥−1 and hence there exists anarithmetical function w∗ : N −→ N such that

Cn

Dn=

pw∗(n)

qw∗(n), n ≥ 1.

It follows from a simple calculation that

µ(Sins) =log(G2/2)

3 log G,

and hence

limn→∞

w∗(n)n

=3 log G

log 2G= 1.2292 · · · a.e.

2.2.3 Approximation coefficients

Approximation coefficients for the OddCF-expansion of an irrational numberx ∈ [0, 1) are defined by

Θn := q2n

∣∣∣∣x−pn

qn

∣∣∣∣ , n ≥ 0. (2.9)

The approximation coefficients (Θn(x))n≥0 of an irrational number x indicatethe quality of approximation of x by its (rational) convergents. For instance,for the RCF the approximation coefficients (θn(x))n≥0 − which are defined by

θn := Q2n

∣∣∣∣x−Pn

Qn

∣∣∣∣ , n ≥ 0,

where (PN/Qn)n≥0 is the sequence of RCF convergents of x − several classicalresults are known. To mention a few results:

28 ODD CONTINUED FRACTIONS

• (Vahlen, 1895) min(θn−1, θn) < 1/2 for all irrational x and n ≥ 1.

• (E. Borel, 1903) min(θn−1, θn, θn+1) < 1/√

5 for all irrational x and n ≥ 1.

• (Legendre) Suppose that p and q > 0 are integers which are relativelyprime, such that q2 |x− p/q| ≤ 1/2. Then p/q is an RCF convergent of x.

For proofs of these results, see e.g., [Sc].

In this section we will obtain a ‘Borel-type’ result for the OddCF. It can beshown (see, e.g., [S2]) that

Θn =(

1Tnx

+en+1qn−1

qn

)−1

=(

en+2Tn+1x +

qn+1

qn

)−1

. (2.10)

Consider the function ψ : Ω −→ R2 defined by

ψ(tn, vn) :=( |vn|

1 + tnvn,

tn1 + tnvn

),

where tn = Tn(x) and vn = en+1qn−1/qn. It follows from (2.10) that

ψ(tn, vn) = (Θn−1, Θn).

Hence, (Θn−1, Θn)n≥1 is distributed over the two regions Γ− in case en+1 = −1and Γ+ in case en+1 = +1, where Γ− is a quadrangle with vertices (0,0), (g2, 0),(g, G), and (0,1) and Γ+ a quadrangle with vertices (0,0), (G, 0), (g, g2), and(0,1); see Figure 2.2.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(i) Γ−: from Schweiger’s natural (ii) Γ+: from Schweiger’s naturalextension with y < 0 extension with y > 0

Figure 2.2: ΓOddCF

Moreover, both Γε with ε ∈ +,− are divided into regions according to thevalue of the partial quotient an. We denote these regions by Γε

a in case an = a.

Insertions, singularizations and the OddCF 29

Note also that for a ≥ 3, each Γεa is divided into two regions according to en+2

by the lines

β = −ec2α + c, c =13,

15,

17, . . . ;

which are the dotted lines in Figure 2.2. We denote these regions by Γεa,+1 in

case en+2 = +1 and Γεa,−1 in case en+2 = −1. For an = 1, however, we only

have Γ+1 and Γ−1 accordingly.

The inverse of ψ is given by

ψ−1(α, β) =(−1 +

√1 + 4αβ

2α,1−√1 + 4αβ

)

for (α, β) ∈ Γ−, and

ψ−1(α, β) =(

1−√1− 4αβ

2α,1−√1− 4αβ

)

for (α, β) ∈ Γ+.Now consider the operator F : ΓOddCF −→ ΓOddCF, where ΓOddCF = Γ+ ∪

Γ−, defined byF := ψT ψ−1.

It follows from the definition of F that

F (α, β) =

(β , e(−α + a√

1 + 4αβ − a2β)), (α, β) ∈ Γ−a,e ,

(β , e(α + a√

1− 4αβ − a2β)), (α, β) ∈ Γ+a,e .

Since F (Θn−1, Θn) = (Θn, Θn+1) we therefore see that

Θn+1 = en+2(en+1Θn−1 + an

√1− 4en+1Θn−1Θn − a2

nΘn). (2.11)

To investigate the points in ΓOddCF, we define

Dε = (α, β) ∈ Γε; α ≥ 1/√

5 , β ≥ 1/√

5, ε = ±.

Clearly, for (Θn−1,Θn) ∈ Γε \Dε one has that

min(Θn−1, Θn) <1√5

.

It follows from (2.11) with an = 1, en+1 = +1 and en+2 = +1 that the functionby which the next Θ is calculated on Γ+

1 is given by

h(α, β) = α +√

1− 4αβ − β

This function attains its maximum value on D+ at (1/√

5, 1/√

5), the maximumvalue being 1/

√5. But then we find that for (Θn−1, Θn) ∈ Γ+

min(Θn−1, Θn, Θn+1) ≤ 1√5.

30 ODD CONTINUED FRACTIONS

Now consider points (Θn−1,Θn) ∈ Γ−. Note that (1/√

5, 1/√

5) − which isthe intersection of the lines β = α and β = G4α − G2 − lies on the boundaryof Γ−3,−1. To see what happens with (Θn−1,Θn) ∈ D−, we divide D− into theregions D−

U := D− ∩ Γ−1 and D−D := D− ∩ Γ−3,−1.

On D−U the next Θ is given by the function

h(α, β) = −α +√

1 + 4αβ − β,

which attains it maximum on the boundary of D−U in the point (1/

√5, (10 +√

5)/20), the maximum being

12− 1

4√

5= 0.388196601 · · · <

1√5,

from which it at once follows that for (Θn−1,Θn) ∈ D−U one has that

min(Θn−1, Θn,Θn+1) ≤ 12− 1

4√

5<

1√5.

Thus we only need to focus our attention to D−D = D−∩Γ−3,−1. This is a triangle

with vertices A : (1/√

5, 1/√

5), B : (1/√

5, (10+√

5)/20), and C : (2g3, g). From(2.11) we see that the next Θ on this region is given by the function

h(α, β) = α− 3√

1 + 4αβ + 9β,

which attains its maximum value on the boundary of D−D. It can be checked that

this maximum value is G and is attained at (2g3, g). Hence for (Θn−1, Θn) ∈ Γ−

we havemin(Θn−1, Θn,Θn+1) ≤ 2g3 = 0.4721359 · · · .

Notice that 2g3 is only slightly larger than the classical value 1/√

5. Takingeverything together we have found a ‘Borel-type theorem’.

However, there is more one can say! In D−D we can find a nested sequence

of triangles Dk, k ≥ 1, such that

FD1 ⊂ D−D, FDk+1 ⊂ Dk, for k ≥ 1,

where Dk has vertices F−k(A) : (1/√

5, 1/√

5), Qk, and F−k(C); see Figure 2.3.Here

Qk =

(1√5,

(F2k

F2k+2

)2 1√5

+F2k

F2k+2

),

where Fn, n ≥ 0, are Fibonacci numbers given by

Fn =1√5

(Gn+1 − (−g)n+1

), n ≥ 0,

with g = (√

5− 1)/2 and G = g + 1.

Insertions, singularizations and the OddCF 31

¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤»»»»»

F−k(A) = A

B

C

ÃÃÃÃ

(((

Q1F−1(C)

Q2F−2(C)

...

···((

QkF−k(C)

...···

Figure 2.3: D−D

Refining the argument from above, we see that for each k ≥ 1, the next Θon Dk is given by the function h(α, β) = α−3

√1 + 4αβ +9β, which attains its

maximum value on the boundary of Dk; its maximal value being F−(k−1)(C),attained at F−k(C).

Since F k(Θn−1, Θn) = (Θn+k−1, Θn+k), it follows that

min(Θn−1,Θn, . . . , Θn+k) ≤ π1(F−k(C)), k ≥ 1,

where π1(x, y) = x is the projection on the first coordinate of (x, y).We have the following result.

Theorem 2.3 Let Θn be given by (2.9). Then

min(Θn−1, Θn, Θn+1, . . . , Θn+k) ≤ 1√5(1 + g4k+2), k = 1, 2, . . . .

Proof. From the definition of F , we see that

F−(k−1)(2g3, g) = (Ckg2k+1, Ck−1g2k−1)

where Ck = 3Ck−1 − Ck−2, k ≥ 2, C0 = 1, and C1 = 2, so that

min(Θn−1, Θn,Θn+1, . . . , Θn+k) ≤ Ckg2k+1, k = 0, 1, 2, . . . . (2.12)

Using the explicit formula for Fibonacci numbers and analysis of differences (seesection 2.5 in [SP] for details), we obtain

Ck =1√5

k∑

j=0

(kj

)(Gj+1 − (−g)j+1) =

1√5(G2k+1 + g2k+1),

32 ODD CONTINUED FRACTIONS

and the result follows from (2.12). 2

Remark 2.3 This result coincides with Tong’s result for NICF, see [T]. Thisis due to the fact that the part of D− which lies below the line β = 4

25α + 25

is exactly the triangle D in [JK] and Tong’s result is a generalization on thistriangle. The following corollary follows directly from Theorem 2.3.

Corollary 2.1 (Schweiger, 1984) For any irrational x the inequality

Θn(x) <1√5

is valid for infinitely many n

In the next section we will see that the grotesque continued fractions can beobtained from the RCF via singularizations and unitary-insertions and similarresults will follow; see Remark 2.4.

2.3 Grotesque continued fractions

It is obvious that the GCF can never be obtained via an algorithms in which ansrcf-insertion is applied; one will always have −1/1 somewhere, which violatesone of the rules of the GCF.

The following theorem gives an algorithm which turns the RCF-expansion ofx ∈ [0, 1) into the GCF-expansion of x. The proof of this theorem also followseasily by inspection, and is therefore omitted.

Theorem 2.4 Let x ∈ [0, 1) with RCF-expansion (2.5), i.e., d0 = 0. Thenstarting from the RCF-expansion (2.5) of x, the following algorithm yields theGCF-expansion of x.

(I) Let m := infn ∈ N; dn is even .(i) If dm+1 > 1, insert 1/1− after dm to obtain

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm − 1), 1/1, −1/(dm+1 + 1), 1/dm+2, · · · ] .

(ii) If dm+1 = 1, let k := infn > m; dn > 1 (k = ∞ is allowed). Nowsingularize in the block of partial quotients

dm+1 = 1, dm+2 = 1, . . . , dk−1 = 1

the last, second from last, etc. partial quotients equal to 1, to arrive at

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm + 1), −1/3, · · · , −1/3| z k−m−2

2 −times

, −1/(dk + 1), 1/dk+1, · · · ] ,

Grotesque continued fractions 33

in case k −m− 1 is odd or k = ∞; in the latter case we find

[ 0; 1/d1, · · · , 1/dm−1, 1/(dm + 1), −1/3, · · · , −1/3, · · · ] .

In case k −m− 1 is even we obtain

[ 0; 1/d1, · · · , 1/dm−1, 1/dm, 1/2, −1/3, · · · , −1/3| z k−m−3

2 −times

, −1/(dk + 1), 1/dk+1, · · · ] .

In this case insert 1/1− after 1/dm to arrive at

[ 0; 1/d1, · · · , 1/(dm − 1), 1/1, −1/3, · · · , −1/3| z k−m−1

2 −times

, −1/(dk + 1), 1/dk+1, · · · ] .

(II) Let m ≥ 1 be the first index in the new unitary-expansion [ c0; e1/c1, · · · ]of x obtained in (I) for which cm is even. Repeat the procedure from (I)to this new unitary-expansion of x with this value of m.

As soon as m = ∞ in (II) we have obtained the GCF-expansion of x.

Remark 2.4 It was shown in [K] that Hurwitz’ singular continued fraction(HSCF) expansion of x is obtained from the RCF-expansion of x by applyingthis ‘new’ step (I)(ii) from Theorem 2.4 to any block of regular partial quotientsequal to 1, which is preceded and followed by a regular partial quotient differentfrom 1 (again this restriction does not apply if the expansion of x starts with1’s, or when the block of 1’s is infinite). Thus we see that the sequence of HSCF-convergents of x forms a subsequence of the sequence of GCF-convergents of x.Due to this, Theorem 2.2 has an obvious analogue for the GCF, which we omithere.

Example 2.2 Applying the ‘new’ step (I)(ii) from Theorem 2.4 to x in Exam-ple 2.1, we obtain

[0; 1/3, 1/1, 1/3, 1/1,−1/7, 1/1,−1/3,−1/3,−1/3,−1/3,−1/6, · · · ].

Note that we had inserted 1/1− after 1/4.

Instead of inserting −1/1, we could also insert 1/1− ‘at the appropriateplace’ in any algorithm in the previous subsection. This leads to another classof maximal expansions with odd partial quotients of x which will be discussedin the next section.

34 ODD CONTINUED FRACTIONS

2.4 Other odd continued fractions

2.4.1 Maximal OddCF’s

Notice that in Theorem 2.2 (I)(ii), bk−m2 c 1’s are singularized, which is a direct

consequence of the fact that the NICF is a maximal S-expansion, see also Section4 in [K]. In case k−m−1 is odd one is forced to singularize the first, third, etc.1 to singularize the maximal number of 1’s in the block.

However, in case k −m − 1 is even there is considerable freedom to choosethe k−m−1

2 1’s which should be singularized (in order to singularize as many1’s as possible), and one could also do the following: singularize in the block ofpartial quotients

dm+1 = 1, dm+2 = 1, . . . , dk−1 = 1k−m−1

2 1’s which are not consecutive (two consecutive regular partial quotientsequal to 1 can never be singularized simultaneously), and then insert −1/1 ‘atthe appropriate place’. For instance, singularizing the second, fourth, sixth, etc.partial quotients equal to 1 yields

[ 0; 1/d1, · · · , 1/dm−1, 1/dm, 1/2,−1/3, · · · ,−1/3︸ ︷︷ ︸k−m−3

2 −times

,−1/(dk + 1), 1/dk+1, · · · ].

In this case we need to insert −1/1 ‘in between’ the two even partial quotientsdm and dm+1 + 1 = 2 at the beginning of the block we just obtained via singu-larizations, to arrive at

[ 0; 1/d1, · · · , 1/(dm + 1),−1/1, 1/1,−1/3, · · · ,−1/3︸ ︷︷ ︸k−m−3

2 −times

,−1/(dk + 1), 1/dk+1, · · · ].

If we apply this new ‘algorithm’ to the RCF-expansion of x we get for almostevery2 x a continued fraction expansion of x with odd partial quotients, whichis different from both the OddCF- and the GCF-expansions of x. In generalsuch an odd expansion is called a maximal expansion with odd partial quotientsof x, since a maximal number of possible singularizations is used to obtain thisexpansion from the RCF-expansion of x.

Example 2.3 Applying this ‘new’ algorithm to x in Example 2.1, we obtain

[0; 1/3, 1/1, 1/5,−1/1, 1/7,−1/1, 1/1,−1/3,−1/3,−1/3,−1/6, · · · ].

Note that we inserted −1/1 after 1/4 in Example 2.1 (i).

Maximal continued fractions with odd partial quotients are only ‘locally dif-ferent’ from one-another. Insertions in (I)(i) in Theorem 2.1 will always be at

2All almost all statements in this paper are with respect to Lebesgue measure λ on [0, 1)

Other odd continued fractions 35

the same place in the RCF-expansion of x if one wants to obtain an SRCF-expansion of x with odd partial quotients, and blocks of 1’s of odd length canbe singularized only in one way if one wants to singularize as many 1’s as pos-sible. Differences can only occur when an even block of RCF partial quotientsequal to 1 must be singularized in (I)(ii). As we saw in the example, in thatcase one always needs an extra insertion.

2.4.2 Non-maximal expansions with odd digits

In the previous sub-section we saw that there exist two classes of maximal con-tinued fraction expansions with odd digits. One class was obtained by inserting−1/1 and the other one by inserting 1/1−, and in both classes one singularizedthe maximal number of 1’s possible. (A third class can be obtained by ‘mixing’both types of insertions.) However, it is not necessary to singularize the maxi-mum amount of 1’s possible, as the following example shows.

Example 2.4 Let x ∈ [0, 1) be as in Example 2.1. We must insert either−1/1 or 1/1− after d3 = 4. In the thus obtained expansion of x, the fifthpartial quotient is even, followed by eight partial quotients equal to 1. Nowsingularizing the first, fourth and seventh 1 yields

[0; 1/3, 1/1, 1/5,−1/1, 1/7,−1/2, 1/2,−1/2, 1/2,−1/2, 1/5, · · · ],

in case we inserted −1/1 after d3 = 4, or

[0; 1/3, 1/1, 1/3, 1/1,−1/9,−1/2, 1/2,−1/2, 1/2,−1/2, 1/5, · · · ],in case we inserted 1/1− after d3 = 4. Now inserting −1/1 resp. 1/1− yields

[0; 1/3, 1/1, 1/5,−1/1, 1/7,−1/3,−1/1, 1/1,−1/3,−1/1, 1/1,−1/3,−1/1, 1/4, · · · ],

resp.

[0; 1/3, 1/1, 1/5,−1/1, 1/9,−1/1, 1/1,−1/3,−1/1, 1/1,−1/3,−1/1, 1/1,−1/6, · · · ].

Instead of singularizing the maximum number of 1’s, in Example 2.4 we singu-larized the minimum number of 1’s in the block. Again this leads to two classesof minimal expansions with odd partial quotients of x, depending on which inser-tion used. Notice, that if the number of 1’s in the block ‘that needs to disappear’equals 3` + i, with i = 0, 1, 2, the minimal number of singularizations needed is` if i = 0, and ` + 1 if i 6= 0.

36 ODD CONTINUED FRACTIONS

Bibliography

[A] Adams, W. W. - On the relationship between the convergents of thenearest integer and regular continued fractions, Math. Comp. 33 (1979),1321-1331.

[B1] Barbolosi, D. - Sur le developpement en fractions continues a quo-tients partiels impairs, Monatsh. Math. 109 (1990), no. 1, 25–37. MR91d:11094

[B2] Barbolosi, D. - Automates et fractions continues, J. Theor. NombresBordeaux 5 (1993), no. 1, 1–22. MR 94k:11079

[G] Goldman, Jay R. – Hurwitz sequences, the Farey process, and gen-eral continued fractions, Adv. in Math. 72 (1988), no. 2, 239–260. MR90b:11065

[I] Ito, Shunji – Algorithms with mediant convergents and their metricaltheory, Osaka J. Math. 26 (1989), no. 3, 557–578. MR 90k:11101

[JK] Jager, H. and C. Kraaikamp. - On the approximation by continued frac-tions, Indag. Math. 51 (1989), no. 3, 289–307. MR 90k:11084

[Ka1] Kalpazidou, S. - On a problem of Gauss-Kuzmin type for continued frac-tion with odd partial quotients, Pacific J. Math. 123 (1986), no. 1, 103–114. MR 87k:11086

[Ka2] Kalpazidou, S. - On the application of dependence with complete connec-tions to the metrical theory of G-continued fractions. Dependence withcomplete connections, Litovsk. Mat. Sb. 27 (1987), no. 1, 68–79. MR88m:11063

[Ka3] Kalpazidou, S. - On the entropy of expansions with odd partial quotients,Probability theory and mathematical statistics, Vol. II (Vilnius, 1985),55–61, VNU Sci. Press, Utrecht, 1987. MR 88k:11050

[K] Kraaikamp, C. - A new class of continued fraction expansions, ActaArith. 57 (1991), no. 1, 1–39. MR 92a:11090

37

38 ODD CONTINUED FRACTIONS

[N] Nakada, H. - Metrical theory for a class of continued fraction trans-formations and their natural extentions, Tokyo J. of Math. 4 (1981),399–429. MR 83k:27875

[P] Petersen, Karl - Ergodic theory, Cambridge Studies in Advanced Math-ematics, 2. Cambridge University Press, Cambridge-New York, 1983.MR 87i:28002

[R1] Rieger, G. J. - Ein Heilbronn-Satz fur Kettenbruche mit ungeraden Teil-nennern, Math. Nachr. 101 (1981), 295–307. MR 83c:10011

[R2] Rieger, G. J. - On the metrical theory of continued fractions with oddpartial quotients. Topics in classical number theory, Vol. I, II (Budapest,1981), 1371–1418, Colloq. Math. Soc. Janos Bolyai, 34, North-Holland,Amsterdam-New York, 1984. MR 86j:11080

[Sc] Schmidt, Wolfgang M. - Diophantine Approximation, Lecture Notes inMathematics 785, Springer-Verlag, Berlin, Heidelberg, New York, 1980.

[S1] Schweiger, F. - Continued fractions with odd and even partial quotients,Arbeitbericht Mathematisches Institut Salzburg 4 (1982), 59–70.

[S2] Schweiger, F. - On the approximation by continued fractions with odd andeven partial quotients, Arbeitbericht Mathematisches Institut Salzburg1-2 (1984), 105–114.

[Se1] Sebe, G.I. - Gauss’ problem for the continued fraction with odd partialquotients, to appear in Monatshefte f. Math.

[Se2] Sebe, G.I. - On Convergence Rate in the Gauss-Kuzmin Problemfor Grotesque Continued Fractions, to appear in Revue RoumaineMath. Pures Appl.

[SP] Sloane, N. J. A. and S. Plouffe. -The Encyclopedia ofInteger Sequences. Academic Press, San Diego, 1995.http://www.research.att.com/˜njas/sequences/index.html

[T] Tong, J. C. - Approximation by nearest integer continued fractions. II.Math. Scand. 74 (1994), no 1, 17–18. MR 95c:11085

Chapter 3

Engel Continued Fractions

3.1 Introduction

Over the last 15 years the ergodic properties of several continued fraction expan-sions have been studied for which the underlying dynamical system is ergodic,but for which no finite invariant measure equivalent to Lebesgue measure exists.Examples of such continued fraction expansions are the ‘backward’ continuedfraction (see [AF]), the ‘continued fraction with even partial quotients’ (see[S3]) and the Farey-shift (see [Leh]). All these (and other) continued fractionexpansions are ergodic, and have a σ-finite, infinite invariant measure. Sincethese continued fractions are closely related to the regular continued fraction(RCF) expansion, their ergodic properties follow from those of the RCF by usingstandard techniques in ergodic theory (see [DK]).

In this paper we introduce a new continued fraction expansion, which wecall—for reasons which will become apparent shortly—the Engel continued frac-tion (ECF) expansion. We will show that this ECF has an underlying dynami-cal system which is ergodic, but that no finite invariant measure equivalent toLebesgue measure exists for the ECF.

As the name suggests, the ECF is a generalization of the classical Engelseries expansion, which is generated by the map S : [0, 1) → [0, 1), given by

S(x) :=(b 1xc+ 1

)(x− 1

b 1xc+ 1

), x 6= 0; S(0) := 0,

where bξc is the largest integer not exceeding ξ, see also Figure 3.1.Using S, one can find a (unique) series expansion of every x ∈ (0, 1), given

by

x =1q1

+1

q1q2+ · · ·+ 1

q1q2 · · · qn+ · · · ,

where qn = qn(x) = b1/Sn−1(x)c + 1, n ≥ 1. In fact it was W. Sierpinski [Si]who first studied these series expansions in 1911.

39

40 ENGEL CONTINUED FRACTIONS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.1: Engel series map S

The metric properties of the Engel series expansion have been studied in aseries of papers by E. Borel [B], P. Levy [L], P. Erdos, A. Renyi and P. Szusz[ERS] and Renyi [R]. In [ERS] it is shown that the random variables X1 =log q1, Xn = log(qn/qn−1) are ‘almost independent‘ and ‘almost identically dis-tributed’. From this first the central limit theorem is derived for log qn, thenthe strong law of large numbers and finally the law of the iterated logarithm areobtained. The second result has been announced earlier without proof by E.Borel [B]. The first and third results are due to P. Levy [L]. In [R], Renyi findsnew (and more elegant) proofs to these and other results. Later F. Schweiger[S1] showed that S is ergodic, and M. Thaler [T] found a whole family of σ-finite,infinite measures for S. Further information on the Engel series (and the relatedSylvester Series) can be found in the books by J. Galambos [G] and Schweiger[S3].

Clearly one can generalize the Engel series expansion by changing the time-zero partition, or by ‘flipping’ the map S on each partition element (thus ob-taining an alternating Engel series expansion, see also [K2K]). Let (rn)n≥1 bea monotonically decreasing sequence of numbers in (0, 1), with r1 = 1, rn > 0for n ≥ 1 and limn→∞ rn = 0. Furthermore, let `n = rn+1, n ≥ 1, and let thetime-zero partition P be given by

P := [`n, rn); n ∈ N .

ThenSE(x) :=

rn

rn − `n(x− `n), x 6= 0; SE(0) := 0,

where n ∈ N is such that x ∈ [`n, rn) generalizes the ‘Engel-map’ S. With someeffort the ideas from the above mentioned papers can be carried over to thisgeneralization, also see W. Vervaat’s thesis [V].

Introduction 41

In this paper we study a different variation of the Engel series expansion.Let the Engel continued fraction (ECF) map TE : [0, 1) → [0, 1) be given by

TE(x) :=1b 1

xc

(1x− b 1

xc)

, x 6= 0; TE(0) := 0, (3.1)

see also Figure 3.2. Notice that

TE(x) =1

a1(x)T (x), 0 ≤ x < 1,

where T : [0, 1] → [0, 1) is the regular continued fraction map, given by

T (x) :=1x− b 1

xc, 0 < x < 1; T (0) := 0,

and a1(x) = b1/xc. For any x ∈ (0, 1), the ECF-map ‘generates’ (in a way

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ECF−mapRCF−map

Figure 3.2: The ECF-map TE and the RCF-map T

similar to the way the RCF-map T ‘generates’ the RCF-expansion of x) a newcontinued fraction expansion of x of the form

1

b1 +b1

b2 +b2

b3 +.. . +

bn−1

bn +.. .

, bn ∈ N, bn ≤ bn+1. (3.2)

42 ENGEL CONTINUED FRACTIONS

In this paper we will study in Section 3.3 the ergodic properties of this newcontinued fraction expansion, the so-called Engel continued fraction (ECF) ex-pansion. However, since the ECF is new, we will first show in the next sectionthat the ECF ‘behaves’ in many ways like any other (semi-regular) continuedfractions. In the last section we will study the relation between the ECF and acontinued fraction expansion introduced by F. Ryde [Ry1] in 1951, the so-calledmonotonen, nicht-abnehmenden Kettenbruch (MNK). We will show that theECF and a minor modification of Ryde’s MNK are metrically isomorphic, anddue to this many properties of the ECF—such as ergodicity, the existence of σ-finite, infinite measures—can be carried over to the MNK. Conversely, the factthat not every quadratic irrational x has an ultimately periodic MNK-expansioncan be carried over to the ECF via our isomorphism.

3.2 Basic properties

In this section we will study the basic properties of the ECF. In many ways itresembles the RCF, but there are also some open questions which suggest thatthere are fundamental differences.

Let x ∈ (0, 1), and define

b1 = b1(x) := b1/xc

bn = bn(x) := b1(Tn−1E (x)), n ≥ 2, Tn−1

E (x) 6= 0.(3.3)

From definition (3.1) of TE it follows that

x =1

b1 + b1TE(x)= · · · = 1

b1 +b1

b2 +b2

b3 +.. . +

bn−1

bn + bnTnE(x)

,

where T 0E(x) = x and Tn

E(x) = TE(Tn−1E (x)) for n ≥ 1, and one has—similar to

the Engel series case—that

1 ≤ b1 ≤ b2 ≤ · · · ≤ bn ≤ · · · . (3.4)

As usual the convergents are obtained via a finite truncation:

1

b1 +b1

b2 +.. . +

bn−1

bn

= Cn, n ≥ 1. (3.5)

The finite continued fraction in the left-hand side of (3.5) is denoted by[[ 0; b1, · · · , bn]].

It is clear that the left-hand side of (3.5) is a rational number, so that wehave the following result.

Basic properties 43

Theorem 3.1 Let x ∈ (0, 1), then x has a finite ECF-expansion (i.e., TnE(x) =

0 for some n ≥ 1) if and only if x ∈ Q.

Proof. The necessary condition is obvious since x in (3.5) is a rational number.For the sufficient condition, let x = P/Q with P, Q ∈ N and 0 < P < Q. Then

TE(x) =

Q

P− bQ

Pc

bQPc

=Q− b1P

b1P,

where b1 = bQ/P c. It is clear that 0 ≤ Q− b1P < P and 1 ≤ b1P ≤ Q becauseQ = b1P + r with 0 ≤ r < P by the Euclidean algorithm. Now let

TnE(x) =

P (n)

Q(n), n = 0, 1, 2, . . .

then0 ≤ · · · < P (n+1) < P (n) < · · · < P (0).

Since P (N) ∈ N ∪ 0 there exists an n ≥ 1 such that TnE(x) = 0. Notice that

one has to apply TE to x at most P times to get TnE(x) = 0. 2

The proof of the following theorem is omitted, since it is quite straightfor-ward. For a proof the interested reader is referred to Section 1 in [K], where asimilar result has been obtained for a general class of continued fractions.

Proposition 3.1 Let the sequences (Pn)n≥0 and (Qn)n≥0 recursively be definedby

P0 = 0, P1 = 1, Pn = bnPn−1 + bn−1Pn−2 for n ≥ 2,

Q0 = 1, Q1 = b1, Qn = bnQn−1 + bn−1Qn−2 for n ≥ 2.(3.6)

Then (Pn)n≥0 and (Qn)n≥0 are both increasing sequences. Furthermore, onehas for n ≥ 1 that

Cn =Pn

Qn, n ≥ 1,

and

PnQn−1 − Pn−1Qn = (−1)n−1n−1∏

j=1

bj . (3.7)

For the RCF-expansion it is known that even-numbered convergents arestrictly increasing and odd-numbered strictly decreasing and that every even-numbered convergent is less than every odd-numbered one. The same resultalso holds for ECF convergents as stated in the following proposition.

Proposition 3.2 Let Cn = Pn/Qn be the n-th ECF-convergent of x ∈ [0, 1)\Q.Then

0 = C0 < C2 < · · · < C3 < C1 ≤ 1.

Moreover, C2j < C2k+1 for any nonnegative j and k.

44 ENGEL CONTINUED FRACTIONS

Proof . It follows from Proposition 3.1, using (3.6), that

PnQn−2 − Pn−2Qn = bn(−1)n−2n−2∏

j=1

bj .

Dividing both sides by QnQn−2 gives

Cn − Cn−2 = (−1)n−2bn

∏n−2j=1 bj

QnQn−2.

Since bj ≥ 1 for all j (and so are the Qj ’s), Cn − Cn−2 > 0 if n is even. Hence,C2m−2 < C2m for m ≥ 1. Similarly, C2m+1 < C2m−1. Now upon division onboth sides of (3.7) by QnQn−1 one obtains

Cn − Cn−1 = (−1)n−1(n−1∏

j=1

bj)/QnQn−1

which gives Cn−1 < Cn if n is odd, that is, C2m < C2m+1. Thus for anynonnegative j and k

C2j < C2j+2k < C2j+2k+1 < C2k+1,

as desired. 2

In the next proposition, we will see that the sequence of ECF-convergentsconverges to the number from which it is generated.

Proposition 3.3 For x ∈ [0, 1), let (Pn/Qn)n≥1 be the sequence of ECF-convergents of x. Then limn→∞ Pn/Qn = x.

Proof . In case x is rational, the result is clear; see Theorem 3.1. Now supposethat x is irrational. By induction one has that

x =Pn + bnTn

E(x)Pn−1

Qn + bnTnE(x)Qn−1

, n ≥ 1. (3.8)

In fact, if x is rational the special case TnE(x) = 0 gives x = Pn/Qn. From (3.7)

and (3.8) it follows that

x− Pn−1

Qn−1=

(−1)n−1∏n−1

j=1 bj

Qn−1(Qn + bnTnE(x)Qn−1)

, n ≥ 2, (3.9)

which trivially yields that

∣∣∣∣ x− Pn

Qn

∣∣∣∣ ≤∏n

j=1 bj

QnQn+1, n ≥ 1. (3.10)

Basic properties 45

Letting φn =∏n

j=1 bj and using (3.6), one can see that φ2n is one of the terms

in QnQn+1 so that the right-hand side of (3.10) goes to zero as n → ∞. Thiscompletes the proof. 2

Notice that (3.9) yields that C2n < x < C2n+1, for n ≥ 1, where Cn =Pn/Qn. We will now show that the ECF-expansion is unique.

Proposition 3.4 Let (bn)n≥1 be a sequence of positive integers satisfying (3.4),and let the sequences (Pn)n≥0 and (Qn)n≥0 be given by (3.6). Then the limit

limn→∞

Pn

Qn

exists. Say this limit equals x, then x ∈ (0, 1). Furthermore, bn = bn(x) forn ≥ 1.

Proof. Let Cn = Pn/Qn. From the proof of Proposition 3.2 it follows that (C2k)is an increasing sequence that is bounded from above; therefore, limk→∞ C2k

exists. Similarly, (C2k+1) is a decreasing sequence that is bounded from below,and so limk→∞ C2k+1 also exists. It remains to show that these two limits areequal. To this end, note that, in the proof of Proposition 3.2, C2k+1 −C2k → 0as k → ∞ (by the same argument as in the proof of Proposition 3.3) so thatlimk→∞ C2k+1 = limk→∞ C2k. Let x = limk→∞ C2k; then limk→∞ Ck existsand equals x. This completes the first part of the proof.

Since 0 < C2 < x < C1 ≤ 1, the second statement that x ∈ (0, 1) followstrivially.

For the third part, recall that by definition of Cn we have for n ≥ 2

Cn = [[ 0; b1, · · · , bn]] =1

b1 + b1[[ 0; b2, · · · , bn]]. (3.11)

Setting for n ≥ 2Cn = [[ 0; b2, · · · , bn]] ,

it follows from the first part of the proof that there exists a number x ∈ (0, 1)such that limn→∞ Cn = x. By letting n →∞ in (3.11) we find that

x =1

b1 + b1x,

from which1x − b1

b1= x ∈ (0, 1) . (3.12)

Since b1(x) is the unique positive integer for which 1x − b1(x) ∈ [0, 1), it follows

that b1(x) is also the unique positive integer for which

TE(x) =1x − b1(x)

b1(x)∈ [0, 1).

46 ENGEL CONTINUED FRACTIONS

But then it follows from (3.12) that b1(x) = b1, and moreover we see thatTE(x) = x. Repeating the above argument, now applied to TE(x), yields thatb2(x) = b2. By induction one finds that bn(x) = bn for n ∈ N. 2

Let x ∈ [0, 1) \Q. We denote the (infinite) ECF-expansion (3.2) of x by

x = [[ 0; b1, b2, · · · , bn, · · · ]] .For the RCF-expansion one has the theorem of Lagrange, which states that x

is a quadratic irrational (i.e., x ∈ R\Q and is a root of ax2 +bx+c = 0, a, b, c ∈Z) if and only if x has a RCF-expansion which is ultimately (that is, from somemoment on) periodic. For the ECF the situation is more complicated. Supposex ∈ [0, 1) has a periodic ECF-expansion, say

x = [[0; b1, · · · , bp, bp+1, · · · , bp+`]],

where the bar indicates the period. Clearly one has that x is a quadratic ir-rational, and from (3.4) it follows that the period-length ` is always equal to1.

Due to this, purely periodic expansions can easily be characterized; for n ∈ None has

τn :=−n +

√n2 + 4n

2n= [[ 0; n]] . (3.13)

One could wonder whether every quadratic irrational x has an eventually peri-odic ECF-expansion. In [Ry2], Ryde showed that a quadratic irrational x hasan eventually periodic MNK-expansion if and only if a certain set of conditionsare satisfied. In Section 3.4 we will see that the ECF-map TE and a mod-ified version of the MNK-map are isomorphic, and due to this we will obtainthat not every quadratic irrational x has an eventually periodic ECF-expansion.

An important question is the relation between the convergents of the RCFand those of the ECF. Let x ∈ [0, 1)\Q, with RCF-expansion x = [ 0; a1, a2, . . . ],with RCF-convergents (pn/qn)n≥1 and ECF-convergents (Pn/Qn)n≥1. More-over, define the mediant convergents of x by

apn + pn−1

aqn + qn−1, for 1 ≤ a ≤ an+1 − 1 ,

then the question arises whether infinitely many RCF-convergents and/or me-diants are among the ECF-convergents, and conversely.

Example 3.1 Let x = 37 + 5√

15142 = 0.396936 · · · , then

x = [ 0; 2, 1, 1, 12, 2, 5, 19, 5, 2 ] = [[ 0; 2, 3, 3, 5, 6 ]]

and P1Q1

= 12 = p1

q1, P2

Q2= 3

8 (a mediant), P3Q3

= 25 = p3

q3, P4

Q4= 23

28 (a mediant), P5Q5

is neither a RCF-convergent nor a mediant, P6Q6

= 181456 (a mediant), P7

Q7= 622

1567 (a

Ergocic properties 47

mediant), P8Q8

= 285718 = p6

q6, etc. Among the first 15 ECF-convergents P5

Q5= 79

199

and P11Q11

= 3855497129 are neither RCF-convergents nor mediants.

A related question is the value of the first point in a ‘Hurwitz spectrum’ forthe ECF. Let x ∈ [0, 1) \Q, again with RCF-convergents (pn/qn)n≥0 and ECF-convergents (Pn/Qn)n≥0, where we moreover assume that gcd(Pn, Qn) = 1 forn ≥ 0. Setting for n ≥ 0

θn = θn(x) := qn|qnx− pn| and Θn = Θn(x) := Qn|Qnx− Pn| ,

one has the classical results that θn < 1 and

min(θn−1, θn, θn+1) <1√

a2n+1 + 4

for n ≥ 1,

which trivially implies that

θn <1√5

infinitely often for all x ∈ [0, 1) \Q.

If infinitely many RCF-convergents of x are also ECF-convergents of x, thenone has that

Θn < C infinitely often, (3.14)

with C = 1. Consider the number x having a purely periodic expansion withbn = 2. Then x = 1

2 (−1 +√

3). The difference equation An = 2An−1 + 2An−2

controls the growth of Θn(x) = Qn|Qnx− Pn|. Its eigenvalues are 1−√3 and1 +

√3, and therefore we see that Θn(x) = Qn|Qnx − Pn| is asymptotically

equal to 2n. Furthermore from (3.9) one can see thatQn

j=1 bj

bn+1+2 ≤ Θn(x) =

Qn|Qnx − Pn| ≤Qn

j=1 bj

bn+1, which shows again that such a constant C cannot

exist for all x.

3.3 Ergodic properties

In this section we will show that TE has no finite invariant measure, equivalentto the Lebesgue measure λ, but that TE has infinitely many σ-finite, infiniteinvariant measures. Furthermore it is shown that TE is ergodic with respect to λ.

Let

B(n) :=[

1n + 1

,1n

)for n ∈ N , (3.15)

and define for n ∈ N, b1, · · · , bn ∈ N with b1 ≤ . . . ≤ bn the cylinder sets (orfundamental intervals) B(b1, . . . , bn) by

B(b1, . . . , bn) =x ∈ [0, 1); T i−1

E x ∈ B(bi), i = 1, . . . , n

.

48 ENGEL CONTINUED FRACTIONS

Then it is clear that, see also Figure 3.2,

T−1E B(n) =

n⋃

k=1

(n

k(n + 1),

n + 1k(n + 2)

]. (3.16)

Now let µ be a finite TE-invariant measure, that is,

µ(T−1E A) = µ(A)

for any Borel set A ∈ [0, 1).Since µ is a measure, we have that

µ([1/2, τ1]) = µ([1/2, 3/5)) + µ([3/5, τ1]),

where τn ∈ B(n) denotes the invariant point under TE , see also (3.13). On theother hand,

µ([1/2, τ1]) = µ(T−1E [1/2, τ1])

= µ([τ1, 2/3])= µ(T−1

E [τ1, 2/3])= µ([3/5, τ1])

because µ is TE-invariant. Hence, since we assumed that µ is a finite measure,

µ([1/2, 3/5)) = 0.

Furthermore, we also have

µ((2/3, 1)) = µ(T−1E (2/3, 1)) = µ((1/2, 3/5)) = 0.

Similar arguments yield that µ([3/5, τ1]) = µ([8/13, τ1]), and therefore

µ((5/8, 2/3]) = µ(T−1E (5/8, 2/3]) = µ([3/5, 8/13)) = 0

so that we have

µ([1/2, 8/13]) = 0 and µ([5/8, 1]) = 0.

Continue this iteration to see that B(1) must have its mass µ(B(1)) concentratedat τ1. Next, it follows from (3.16) that T−1

E B(2) = (1/3, 3/8] ∪ (2/3, 3/4]. Butµ(2/3, 3/4] = 0 so that µ(T−1

E B(2)) = µ((1/3, 3/8]) and following the samearguments as above gives that B(2) has its mass µ(B(2)) concentrated at τ2.

Applied to other values of n, induction yields that on (0, 1) the measure µhas mass µ(B(n)) concentrated at τn for n ≥ 1. Consequently, we have provedthe following result.

Theorem 3.2 There does not exist a non-atomic finite TE-invariant measure.

Next, we will prove ergodicity of TE with respect to Lebesgue measure.

Ergocic properties 49

Theorem 3.3 TE is ergodic with respect to Lebesgue measure λ.

Proof. Let T−1E A = A be an invariant Borel set. Define d(b) := b

∫ 1b

0cA(y)dy,

where cA denotes the indicator function of A. Then we calculate

d(b)− d(b + 1) = b

∫ 1b

0

cA(y)dy − (b + 1)∫ 1

b+1

0

cA(y)dy

= b

∫ 1b

1b+1

cA(y)dy −∫ 1

b+1

0

cA(y)dy

=1

b + 1(b(b + 1)

∫ 1b

1b+1

cA(y)dy − (b + 1)∫ 1

b+1

0

cA(y)dy).

We put

δ(b1, ..., bn) :=λ(A ∩B(b1, ..., bn))

λ(B(b1, ..., bn))=

λ(T−nA ∩B(b1, ..., bn))λ(B(b1, ..., bn))

=∫ 1

bn0 ω(b1, ..., bn; y)cA(y)dy

∫ 1bn

0 ω(b1, ..., bn; y)dy.

Note that TnEB(b1, ..., bn) = [0, 1

bn), and that it follows from (3.8) and (3.7) that

ω(b1, ..., bn; y) =∏n

i=1 bi

(Qn + Qn−1bny)2, with 0 ≤ y ≤ 1

bn.

From this it follows that

λ(B(b1, ..., bn)) =

∏n−1j=1 bj

Qn(Qn + Qn−1). (3.17)

Moreover we find that

δ(b) = b(b + 1)∫ 1

b

1b+1

cA(y)dy , (3.18)

and that ∏ni=1 bi

4Q2n

≤ ω(b1, ..., bn; y) ≤∏n

i=1 bi

Q2n

,

which shows that

(A) d(bn)4 ≤ δ(b1, ..., bn) ≤ 4d(bn).

For n = 1 we have a more precise estimate. Since ω(b; y) = 1b(1 + y)2

for

0 ≤ y ≤ 1b

, we get

b

(b + 1)2≤ ω(b; y) ≤ 1

b, for 0 ≤ y ≤ 1

b.

Together with (3.18) this yields that

50 ENGEL CONTINUED FRACTIONS

(B) b(b + 1) d(b) ≤ δ(b) ≤ (b + 1)

bd(b).

Furthermore we have that

(C) d(b)− d(b + 1) = δ(b)− d(b + 1)b + 1 .

Note that for Engel’s series δ(b) = d(b) which fact makes the proof easier.

Together with (B) we get the estimate d(b + 1) ≤ b2 + b + 1b2 + b

d(b).

Setting∏∞

b=1b2+b+1

b2+b =: γ we get

(D) d(c) ≤ γd(b) for all c ≥ b.

The Martingale Convergence Theorem shows that

limn→∞

δ(b1(x), ..., bn(x)) = cA(x) almost everywhere,

see also Theorem 9.3.3 in [S3]. If cA(x) = 1 a.e. there is nothing to show. Let ustherefore assume that λ(A) < 1. Suppose that limn→∞ δ(b1(z), ..., bn(z)) = 0,for some z ∈ Ac. Then for n sufficiently large δ(b1(z), ..., bn(z)) ≤ ε

16γ for anygiven ε > 0, and by (A) for b = bn(z) we find d(b) ≤ ε

4γ . Applying (D) thisyields that d(c) ≤ ε

4 for all c ≥ b.Since the set FN = x : bj(x) ≤ N, j ≥ 1 is countable (since every x ∈ FN

ends in a periodic ECF-expansion with digit b ≤ N), it follows that FN hasmeasure 0 and we clearly have limn→∞ bn = ∞ a.e. Therefore by (A) we seethat δ(b1(x), ..., bn(x)) ≤ ε for almost all points x. Assuming that ε < 1 thisshows that cA(x) = 0 almost everywhere, i.e., λ(A) = 0. 2

For the Engel’s series Renyi [R] showed that for almost all x the sequence ofdigits is monotonically increasing from some moment n0(x) on. For the ECF asimilar result holds.

Theorem 3.4 For almost all x ∈ [0, 1) the sequence of digits (bn(x))∞n=1 isstrictly increasing for some n ≥ n0(x).

Proof. Setting y := Qn−1Qn

, it follows from (3.17) and (3.6) that

λ(B(b1, ..., bn, bn+1))λ(B(b1, ..., bn))

=bn(1 + y)

(bn+1 + bny)(bn+1 + 1 + bny), (3.19)

and that 0 ≤ y ≤ 1bn

.If we put bn = bn+1, we immediately get

λ(x : bn(x) = bn+1(x)) ≤∑

b1≤...≤bn

λ(B(b1, ..., bn))bn + 1

, n ≥ 1.

Ergocic properties 51

Lemma 3.1∑

b1≤...≤bn

λ(B(b1, ..., bn))bn + 1 ≤ ( 313

324 )n .

Proof (of the lemma, see also [S3], pp. 68-69). It follows from (3.19) that

b1≤...≤bn+1

λ(B(b1, ..., bn+1))bn+1 + 1

equals

b1≤...≤bn

λ(B(b1, ..., bn))bn + 1

bn≤bn+1

(bn + 1)bn(1 + y)(bn+1 + 1)(bn+1 + bny)(bn+1 + 1 + bny)

.

Therefore we have to estimate the sum∞∑

b=a

(a + 1)a(1 + y)(b + 1)(b + ay)(b + 1 + ay)

.

For b = a the first term gives 1a+1+ay ≤ 1

a+1 , while for b ≥ a+1 we use 0 ≤ y ≤ 1a

to obtain the estimate

(a + 1)a(1 + y)(b + 1)(b + ay)(b + 1 + ay)

≤ (a + 1)2

b(b + 1)2,

and it follows that∞∑

b=a

(a + 1)a(1 + y)(b + 1)(b + ay)(b + 1 + ay)

≤ 1a + 1

+∞∑

b=a+1

(a + 1)2

b(b + 1)2

≤ 1a + 1

+ (a + 1)2(

1(a + 1)(a + 2)2

+1

(a + 2)(a + 3)2+

∫ ∞

a+2

dz

z(z + 1)2

).

Using∫ ∞

a+2

dz

z(z + 1)2= log

a + 3a + 2

− 1a + 3

≤ 1(a + 2)(a + 3)

− 12(a + 2)2

+1

3(a + 2)3,

we eventually get

∞∑

b=a

(a + 1)a(1 + y)(b + 1)(b + ay)(b + 1 + ay)

≤ 1a + 1

+ (a + 1)2( 1(a + 1)(a + 2)2

+1

(a + 2)(a + 3)2+

1(a + 2)(a + 3)

− 12(a + 2)2

+1

3(a + 2)3).

This expression has its maximum for a = 1 which gives the value 313324 .

The claim on the maximal value can be seen as follows.The sum of the four terms

1a + 1

+a + 1

(a + 2)2+

(a + 1)2

(a + 2)(a + 3)2+

(a + 1)2

3(a + 2)3

52 ENGEL CONTINUED FRACTIONS

decreases to 0 as a → ∞ and becomes smaller than 12 for a ≥ 3. On the other

hand, the remaining term

(a + 1)2( 1(a + 2)(a + 3)

− 12(a + 2)2

)=

(a + 1)3

2(a + 2)2(a + 3)

is increasing and is bounded by 12 . Therefore numerical calculations suffice for

n = 1, 2, 3. This proves the Lemma.

Now the Borel-Cantelli lemma yields that the set of all points x for whichbn(x) = bn+1(x) for infinitely many values of n has measure 0. 2

Now we give two constructions of σ-finite, infinite invariant measures for TE .The first construction follows to some extent Thaler’s construction from [T] ofσ-finite, infinite invariant measures for the Engel’s series map S.

First, let B(n) be as in (3.15). Define

A(n, k) = x; bj = n, 1 ≤ j ≤ k, bk+1 > n .

Obviously one has that A(n, k)∩A(m, `) = ∅ for (n, k) 6= (m, `) and that, apartfrom a set of Lebesgue measure zero

B(n) =∞⋃

k=1

A(n, k).

We now choose a monotonically increasing sequence of positive real numbers(an)∞n=1, satisfying

an >n

n− 1

n−1∑

j=1

aj

j, n ≥ 2.

For any non-purely periodic x ∈ B =⋃∞

n=1 B(n), there exist positive integersn and ` such that x ∈ A(n, `). For any non-eventually periodic x ∈ B \ Q weinductively define a sequence (αk(x))∞k=1 by

α1(x) = an if x ∈ A(n, `) for some ` ≥ 1,

and

αk+1(x) = n (TE(x) + 1)2 αk(TE(x))− n

n−1∑

j=1

aj

j, k ≥ 1.

For x ∈ [0, 1) \Q given one has, since α1(x) = an ≤ α1(TE(x)),

α2(x)− α1(x) ≥ (n(TE(x) + 1)2 − 1)an − n

n−1∑

j=1

aj

j

> n

(n (TE(x) + 1)2 − 1

n− 1− 1

)n−1∑

j=1

aj

j

> 0,

Ergocic properties 53

and for k ≥ 2 one has

αk+1(x)− αk(x) = n(TE(x) + 1)2 (αk (TE(x))− αk−1 (TE(x))) .

So by induction it follows that the sequence (αk(x))∞k=1 is a positive monotoni-cally non-decreasing sequence for each x ∈ [0, 1). We will show that

h(x) := αk(x) for x ∈ A(n, k)

and h(x) := 0 for x 6∈ ∪∞n=1B(n), is a density of a measure equivalent to Lebesguemeasure, that is, it satisfies Kuzmin’s equation, see also Chapter 13 in [S3]. Firstnote that for x ∈ B(n) Kuzmin’s equation reduces to

h(x) =n∑

j=1

h(Vj(x))j(x + 1)2

,

where Vj is the local inverse of TE on B(j), i.e.,

Vj(x) =1

j(1 + x).

Note also that Vn(x) ∈ A(n, k + 1) and Vj(x) ∈ A(j, 1), 1 ≤ j ≤ n − 1, whenx ∈ A(n, k). Therefore, h(x) = αk(x), h(Vn(x)) = αk+1(Vn(x)) and h(Vj(x)) =α1(Vj(x)) = aj , 1 ≤ j ≤ n− 1. We now see that

n∑

j=1

h(Vj(x))j(x + 1)2

=αk+1(Vn(x))n(x + 1)2

+n−1∑

j=1

aj

j(x + 1)2= αk(x) = h(x).

As in Thaler’s case for the Engel series expansion, this construction yields in-finitely many different σ-finite, infinite invariant measures which are not multi-ples of one-another.

For the second construction, let wst (x) =

∣∣∣∣dV s

t (x)dx

∣∣∣∣. Note that w1t (x) =

1t(1 + x)2

. Furthermore, let G > 1 be the golden mean, defined by G2 = G + 1,

g = 1G and

g0(x) :=∣∣∣∣

1(x− g)(x + G)

∣∣∣∣ , 0 ≤ x < 1, x 6= g.

Here f0(x) = x, f1(x) = f(x), fs+1(x) = fs(f(x)). Note that g0 is a solution ofthe functional equation f(x) = f(V1(x))w1

1(x).Setting for t = 2, 3, . . .

gt−1(x) :=∞∑

s=0

gt−2(V st (x))ws

t (x), 0 ≤ x < 1/t,

54 ENGEL CONTINUED FRACTIONS

then h(x) := gt−1(x) on B(t) is an invariant density. To see this note thatKuzmin’s equation reads

h(x) =k∑

s=1

h(Vs(x))w1s(x)

on B(k). Then for x ∈ B(k) we have gk−1(x) on the left hand, while expandingthe right-hand side gives

gk−1(x) = h(V1(x))w11(x) + h(V2(x))w1

2(x) + · · ·+ h(Vk(x))w1k(x)

= g0(V1(x))w11(x) + g1(V2(x))w1

2(x) + · · ·+ gk−1(Vk(x))w1k(x)

= g0(x) +∞∑

s=1

g0(V s2 (x))ws

2(x) + · · ·+∞∑

s=1

gk−2(V sk (x))ws

t (x)

= gk−1(x).

In the calculation we used that ws+1t (x) = ws

t (Vt(x))w1t (x).

We end this section with a theorem on the renormalization of the ECF-mapTE . See also Hubert and Lacroix [HL] for a recent survey of the ideas behindthe renormalization of algorithms.

Settingzn(x) = bn(x)Tn

Ex,

then clearly 0 ≤ zn(x) ≤ 1. We have the following theorem.

Theorem 3.5 Let γ = 313324 , then

λ(x : zn(x) < t) = t(1 + O(γn)) .

Proof. We introduce for t ∈ [0, 1] the map S(b1, ..., bn)(t) := V (b1, ..., bn)( tbn

).Applying the chain-rule we find, see also (3.8),

S′(b1, ..., bn)(t) = (−1)n

∏n−1j=1 bj

(Qn + Qn−1t)2.

Then λ(x : zn(x) < t) =∫ t

0ρn(s)ds, where

ρn(s) = (−1)n∑

b1≤...≤bn

S′(b1, ..., bn)(s) =∑

b1≤...≤bn

∏n−1j=1 bj

(Qn + Qn−1s)2.

On the other hand we have, see also (3.17),

1 =∑

b1≤...≤bn

λ(B(b1, ..., bn)) =∑

b1≤...≤bn

∏n−1j=1 bj

Qn(Qn + Qn−1).

On Ryde’s continued fraction 55

Therefore

|ρn(s)− 1| ≤∑

b1≤...≤bn

∏n−1j=1 bj

Qn(Qn + Qn−1)

∣∣∣∣QnQn−1 − 2sQnQn−1 − s2Q2

n−1

(Qn + Qn−1s)2

∣∣∣∣

≤ 4∑

b1≤...≤bn

λ(B(b1, ..., bn))bn

,

which yields that

ρn(s) = 1 + O(γn) and∫ t

0

ρn(s)ds = t + tO(γn).

2

Remark 3.1 Following the ideas in Schweiger [S2] it should be easy to provethat the sequence (zn(x)) is uniformly distributed for almost all points x.

3.4 On Ryde’s continued fraction with non de-creasing digits

In 1951, Ryde [Ry1] showed that every x ∈ (0, 1) can be written as a monotonen,nicht-abnehmenden Kettenbruch (MNK) of the form

c1

sc1 +c2

c2 +c3

c3 +.. . +

cn

cn +.. .

, cn ∈ N, cn ≤ cn+1, s = b 1xc , (3.20)

which is finite if and only if x is rational.Underlying this expansion is the map SR : (0, 1) → (0, 1), defined by

SR(x) :=

⌊1

1x − b 1

xc

x−

⌊1x

⌊1

1x − b 1

xc

⌋⌋, x ∈ (0, 1).

However, and this was already observed by Ryde, one has that

SR

((0,

12))

=(

12, 1

)= SR

((12, 1)

)(mod 0),

and therefore we might as well restrict our attention to the interval ( 12 , 1), and

just consider the map TR : ( 12 , 1) → ( 1

2 , 1), given by

TR(x) = SR(x) =k

x− k, for x ∈ R(k) :=

(k

k + 1,k + 1k + 2

), k ∈ N,

56 ENGEL CONTINUED FRACTIONS

see also [S3], p. 26. Now every x ∈ ( 12 , 1) has a unique NMK of the form (3.20)

(with s = 1), which we abbreviate by

x =< 0; c1, c2, · · · , cn, · · · > .

The following theorem establishes the relation between the ECF and the MNK.We omitted the proof, since it follows by direct verification.

Theorem 3.6 Let the bijection φ : (0, 1) → ( 12 , 1) be defined by

φ(x) :=1

1 + x, for x ∈ (0, 1).

ThenTR(φ(x)) = φ(TE(x)) , for x ∈ (0, 1). (3.21)

Furthermore, for b1 ≤ b2 ≤ · · · ≤ bn ≤ · · ·

φ([[ 0; b1, b2, · · · , bn, · · · ]]) = < 0; b1, b2, · · · , bn, · · · > ,

and if we define for n ≥ 1 the cylinders of TR by

R(c1, . . . , cn) := x ∈ (12, 1); T i−1

R (x) ∈ R(ci), i = 1, . . . , n , (3.22)

thenφ(B(b1, · · · , bn)) = R(b1, · · · , bn) .

Due to Theorem 3.6 we can ‘carry-over’ the whole ‘metrical structure’ of theECF to the MNK. To be more precise, letting B be the collection of Borel setsof ( 1

2 , 1), and setting

ν(A) := ρ(φ−1(A)) , A ∈ B, (3.23)

where ρ is a σ-finite, infinite TE-invariant measure on (0, 1) with density h (withh from Section 3.3), we have the following corollary.

Corollary 3.1 The map TR is ergodic with respect to Lebesgue measure λ, butno finite TR-invariant measure exists equivalent to λ. Each of the measures νfrom (3.23) is a σ-finite, infinite TR-invariant measure on ( 1

2 , 1).

Proof. We only give a proof of the first statement. Suppose that there exists aBorel set A ⊂ ( 1

2 , 1) for which 0 < λ(A) < λ(12 , 1) = 1

2 , such that T−1R (A) = A.

From the fact that φ : (0, 1) → ( 12 , 1) is a bijection, and due to (3.21) one has

that φ−1(A) is a TE-invariant set, and hence λ(φ−1(A)) ∈ 0, 1, which is im-possible. 2

To conclude this paper, let us return to the question of periodicity of theECF-expansion of a quadratic irrational x. Due to (3.22) we have that the

On Ryde’s continued fraction 57

ECF-expansion of x ∈ (0, 1) is (ultimately) periodic if and only if the NMK-expansion of ξ = φ(x) ∈ ( 1

2 , 1) is (ultimately) periodic. The main result ofRyde’s second 1951 paper [Ry2] now states that a quadratic irrational ξ ∈ (0, 1)has an (ultimately) periodic NMK-expansion if and only if a (rather large) set ofconstraints—too large to be mentioned here; the statement of his theorem coversalmost 2 pages!—has been satisfied. Due to Theorem 3.6 these constraints cantrivially be translated into a set of constraints for the ECF.

As a consequence there exist (infinitely many) quadratic irrationals x forwhich the ECF-expansion is not ultimately periodic. We end this paper withan example.

Example 3.2 Let x = 15 (−1+2

√5) = 0.6944271 · · · . Then the RCF-expansion

of x is x = [ 0; 1, 2, 3, 1, 2, 44, 2, 1, 3, 2, 1, 1, 10, 1 ]. Using MAPLE we ob-tained the first 22 partial quotients of the ECF-expansion of x:

x = [[ 0; 1, 2, 7, 20, 28, 30, 187, 541, 711, 989, 2280, 7630, 8683, 13941, 26110,

32685, 199856, 866227, 5897902, 8834278, 24269774, 96660239, · · · ]] .

Acknowledgements. We would like to thank the referee for several helpfulcomments and suggestions.

58 ENGEL CONTINUED FRACTIONS

Bibliography

[AF] Adler, Roy L. and Flatto, Leopold – The backward continued fractionmap and geodesic flow, Ergodic Theory Dynamical Systems 4 (1984),no. 4, 487–492. MR 86h:58116

[B] Borel, E. – Sur les developpements unitaires normaux, C. R. Acad. Sci.Paris 225, (1947). 51. MR 9,292c

[DK] Dajani, K. and Kraaikamp, C. –The Mother of All Continued Fractions,Coll. Math. 84/85 (2000), 109-123.

[ERS] Erdos, P., Renyi, A. and Szusz, P. – On Engel’s and Sylvester’s series,Ann. Univ. Sci. Budapest. Etvs. Sect. Math. 1 1958 7–32. MR 21 #1288

[G] Galambos, J. – Representations of real numbers by infinite series, Lec-ture Notes in Mathematics, Vol. 502. Springer-Verlag, Berlin-New York,1976. MR 58 #27873

[HL] Hubert, P. and Lacroix, – Renormalization of algorithms in the proba-bilistic sense, New trends in probability and statistics, Vol. 4 (Palanga,1996), 401–412, VSP, Utrecht, 1997. MR 2000c:11131

[K] Kraaikamp, C. – A new class of continued fraction expansions, ActaArith. 57 (1991), no. 1, 1–39. MR 92a:11090

[K2K] Kalpazidou, S., Knopfmacher, A. and Knopfmacher, J. – Luroth-type al-ternating series representations for real numbers, Acta Arith. 55 (1990),no. 4, 311–322. MR 91i:11011

[Leh] Lehner, Joseph – Semiregular continued fractions whose partial denomi-nators are 1 or 2, Contemp. Math., 169 (1994), 407–410, MR 95e:11011

[L] Levy, P. – Remarques sur un theoreme de M. Emile Borel, C. R. Acad.Sci. Paris 225, (1947), 918–919. MR 9,292d

[R] Renyi, A. – A new approach to the theory of Engel’s series, Ann. Univ.Sci. Budapest. Etvs Sect. Math. 5 1962. MR 27 #126 25–32.

[Ry1] Ryde, F. – Eine neue Art monotoner Kettenbruchentwicklungen, Ark.Mat. 1, (1951). 319–339. MR 13,115c

59

60 ENGEL CONTINUED FRACTIONS

[Ry2] Ryde, F. – Sur les fractions continues monotones nondecroissantesperiodiques, Ark. Mat. 1, (1951). 409–420. MR 13,115d

[Si] Sierpinski, W. – Sur quelques algorithmes pour developper les nombresreels en series, In: Oeuvres choisies Tome I, Warszawa 1974, 236–254.MR 54 #2405

[S1] Schweiger, F. – Ergodische Theorie der Engelschen und SylvesterschenReihen, Czechoslovak Math. J. 20 (95) 1970, 243–245. MR 41 #3712;Czechoslovak Math. J. 21 (96) 1971, 165. MR 43 #2190

[S2] Schweiger, F. – Metrische Ergebnisse uber den Kotangensalgorithmus,Acta Arithmetica 26 (1975), 217–22. MR 51 #10269

[S3] Schweiger, F. – Ergodic theory of fibred systems and metric number the-ory, Oxford Science Publications. The Clarendon Press, Oxford Univer-sity Press, New York, 1995. MR 97h:11083

[T] Thaler, M. – σ-endliche invariante Masse fur die Engelschen Reihen,Anz. Osterreich. Akad. Wiss. Math.-Natur. Kl. 116 (1979), no. 2, 46–48. MR 80j:28028

[V] Vervaat, W. – Success epochs in Bernoulli trials (with applications innumber theory), Mathematical Centre Tracts, 42. Mathematisch Cen-trum, Amsterdam, 1972. iii+166 pp. MR 48 #7331

Chapter 4

Tong’s Spectrum for SRCFExpansions

4.1 Introduction

A classical result in Number Theory by Legendre states that irrational numbersx are well approximated only by those rationals, which are continued fractionconvergents of x. To be more precise, let x be an irrational number, with regularcontinued fraction (RCF) expansion

x = [a0; a1, a2, · · · ], (4.1)

where a0 ∈ Z is such that x−a0 ∈ [0, 1), ai ∈ N, i = 1, 2, . . ., and let (pn/qn)n≥−1

be the sequence of regular convergents of x. Then Legendre’s result states thatif p, q ∈ Z, q > 0, and gcd(p, q) = 1, one has that

∣∣∣∣ x− p

q

∣∣∣∣ <12

1q2

⇒ there exists an n ≥ −1 such that(

p

q

)=

(pn

qn

).

In 1891 Hurwitz showed, that for every irrational x

∣∣∣∣ x− p

q

∣∣∣∣ <1√5

1q2

(4.2)

has infinitely many rational solutions p/q, and that the constant in (4.2) is bestpossible, in the sense that it cannot be replaced by a smaller constant. In viewof Legendre’s result each of these solutions is an RCF-convergent of x. Thereare several ways to prove Hurwitz’ result, e.g., Borel showed in 1903 that for allirrational numbers x and all positive integers n one has that

min (θn−1, θn, θn+1) <1√5,

61

62 TONG’S SPECTRUM FOR SRCF EXPANSIONS

where

θk = θk(x) := q2n

∣∣∣∣ x− pk

qk

∣∣∣∣ , k ≥ 0.

The numbers θn, n ≥ 0, are called the RCF-approximation coefficients of x.Typically, that is for almost every x with respect to Lebesgue measure λ,

Hurwitz’ statement can be improved enormously. For almost every x and forall C > 0 one has that ∣∣∣∣ x− p

q

∣∣∣∣ <C

q2(4.3)

has infinitely many rational solutions p/q. This follows from the Doeblin-Lenstraconjecture—proved by W. Bosma, H. Jager and F. Wiedijk in 1983 in [BJW]—which states that for almost every x and every z ∈ [0, 1] the limit

limn→∞

1n

#1 ≤ j ≤ n : θj(x) ≤ z

exists, and equals the distribution function F (z), given by

F (z) =

z

log 2, 0 ≤ z ≤ 1

2 ,

1log 2

(1− z + log 2z) , 12 ≤ z ≤ 1.

So we see that for 0 < C < 12 , for almost every x the fraction of approximation

coefficients smaller or equal to C is C/ log 2, which implies not only that there areinfinitely many rationals p/q satisfying (4.3), but even with a positive density.

In the null-set of numbers x for which the constant 1/√

5 in (4.2) can-not be improved, the numbers x which are equivalent to the golden meang = 1

2

(−1 +√

5)

= 0.61 . . . play an important role. We say that an irrationalnumber x is equivalent to g if there exists a positive integer nx such that an = 1,for all n ≥ nx. If we keep the golden mean g and all its equivalent numbers xout of consideration, we can lower the constant 1/

√5 in (4.2) to 1/

√8. These

two points are the first two points in the so-called Markoff spectrum. See [KS]for a geometrical approach to continued fractions to find these first two point inthe Markoff spectrum, and [CF] or [B] for further information on this spectrum,and the related Lagrange spectrum.

Several (semi-regular) continued fraction algorithms exist for which the approx-imation properties are superior to that of the RCF. In general a semi-regularcontinued fraction algorithm is a finite or infinite continued fraction expansionof the form

b0 +ε1

b1 +ε2

b2 +.. . +

εn

bn +.. .

, (4.4)

Introduction 63

where b0 ∈ Z, bn ∈ N, εn = ±1, and εn+1 + bn ≥ 1, n ≥ 1. In the infinitecase we moreover have that εn+1 + bn ≥ 2 infinitely often. We denote (4.4) by[ b0; ε1/b1, ε2/b2, · · · , εn/bn, · · · ]. For example, if

x = [ b0; e1/b1, e2/b2, · · · , en/bn, · · · ], (4.5)

with b0 ∈ Z, x − b0 ∈ [− 12 , 1

2 ), en = ±1, bn ∈ N and en+1 + bn ≥ 2 for n ≥ 1,then (4.5) is the nearest integer continued fraction (NICF) expansion of x. TheNICF-convergents of x are denoted by

rk

sk= [ b0; e1/b1, e2/b2, · · · , ek/bk], k ≥ 0,

and the NICF-approximation coefficients by

Θk = Θk(x) = s2k

∣∣∣∣ x− rk

sk

∣∣∣∣ , k ≥ 0.

One can show—in fact we will see this below—that the sequence of NICF-convergents of x is a subsequence of the RCF sequence, i.e., there exists afunction n : N→ N which is monotonically increasing such that

(rk

sk

)=

(pn(k)

qn(k)

), k ≥ 1.

For almost all x one has that, setting G = g+1 = 12

(1 +

√5)

(see [A], [J], [K2])

limk→∞

n(k)k

=log 2log G

= 1.44404 · · ·

and (see [BJW])

limk→∞

1k

k∑

i=1

Θi(x) =√

5− 22 log G

= 0.24528 · · · (NICF)

limk→∞

1k

k∑

i=1

θi(x) =1

4 log 2= 0.36067 · · · . (RCF)

We see that the NICF is both faster and closer than the RCF.

In spite of the fact that the approximation properties of the NICF are muchbetter that those of the RCF, Jager and Kraaikamp [JK] obtained in 1989 thesurprising result that for all irrational numbers x and all positive integers n onehas

min (Θn−1,Θn, Θn+1) <52

(5√

5− 11)

= 0.4508 · · · ,

where the constant 52

(5√

5− 11)

(which is larger than 1/√

5 = 0.4472 · · · ) can-not be replaced by a smaller constant. They also showed that for every irra-tional x there are infinitely many positive integers n for which Θn < 1/

√5.

These results were strongly improved by Jingcheng Tong in [T1], [T2], whereTong showed that there exists a ‘pre-spectrum’ to the Markoff spectrum.

64 TONG’S SPECTRUM FOR SRCF EXPANSIONS

Theorem 4.1 (Tong, 1994) For every irrational number x and all positive in-tegers n and k one has

min(Θn−1, Θn, . . . , Θn+k) <1√5

+1√5

(3−√5

2

)2k+3

.

The constant τk(12 ) = 1√

5+ 1√

5

(3−√5

2

)2k+3

is best possible.

In this paper we will show how Tong’s result easily follows from the geometricalapproach from [JK]. This approach also enables us to make several metricobservations on Tong’s pre-spectrum. Moreover, it is shown that this approachworks for a large class of semi-regular continued fractions. For example, aTong spectrum is obtained for every α-expansion of Nakada, and it is alsoexplained why continued fraction algorithms like Bosma’s optimal continuedfraction expansion and Minkowski’s diagonal continued fraction expansion donot have a Tong pre-spectrum.

4.2 Tong’s pre-spectrum for the NICF

We start by recalling some known results on the NICF expansion. Let

Ω = [−12, 0]× [0, g2] ∪ [0,

12)× [0, g],

see Figure 4.1, and define the operator S : Ω → Ω by

− 12 0

g2

g

12

................................................................................................................................

..............................................................................................................

...............................................................................................

...................................................................................

..........................................................................

....................................................

.................................................................................................................................................................................................................................................................................

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

.......

.....

.....

......

......

........

............

..............

Figure 4.1: The natural extension of the NICF and the curves v = C1−Ct and

v = |t|−CCt for C = 0.4.

S(x, y) =(

Sx,1

b1 + e1y

),

Tong’s pre-spectrum for the NICF 65

withSx =

e1

x− b1, S(0) = 0,

where e1 = e1(x) = sgn(x), b1 = b1(x) = b|1/x| + 1/2c, and bξc denotes thelargest integer not exceeding ξ. Setting bn = b1(Sn−1(x)) and en = e1(Sn−1(x)),we see that repeated application of the map S yields the NICF-expansion of anyx ∈ [− 1

2 , 12 ) \ 0. Nakada showed in [Na] that

(Ω,B, µ, S

)forms an ergodic system,

where B is the collection of Borel sets of Ω, and µ is an S-invariant probabilitymeasure on (Ω,B) with density d(x, y) = 0 for (x, y) outside Ω, and

d(x, y) =1

log G

1(1 + xy)2

for (x, y) ∈ Ω.

The system(Ω,B, µ, S

)is called the natural extension of the NICF-expansion.

It is easy to see that

Sn(x, 0) = (Tn, Vn), n = 1, 2, . . . ,

where

Tn := Snx = S(Sn−1x) = [0; en+1/bn+1, en+2/bn+2, · · · ]Vn := sn−1/sn = [0; 1/bn, en/bn−1, · · · , e2/b1].

A direct consequence of Nakada’s result, is that for almost all x the two-dimensional sequence (Tn, Vn) is distributed over Ω according to the densityfunction d(x, y), see [K1]. Furthermore, it can be shown that

Θn−1 =Vn

1 + TnVnand Θn =

en+1Tn

1 + TnVn, (4.6)

see [K1], [JK] or [T2] for details. Since

Tn+1 =en+1

Tn− bn+1 and Vn+1 =

1bn+1 + en+1Vn

,

it follows from the second equality in (4.6) (with n replaced by n + 1) that

Θn+1 =en+2(en+1 − bn+1Tn)(en+1bn+1 + Vn)

1 + TnVn. (4.7)

From (4.6) and (4.7) it follows that

Θn+1 = en+2

(en+1Θn−1 + bn+1

√1− 4en+1Θn−1Θn − b2

n+1Θn

), n ≥ 1,

see also [JK], and

Θn−1 = en+1

(en+2Θn+1 + bn+1

√1− 4en+2ΘnΘn+1 − b2

n+1Θn

), n ≥ 1.

66 TONG’S SPECTRUM FOR SRCF EXPANSIONS

For Tn > 0 we see that for C > 0

Θn−1 ≤ C ⇔ Vn ≤ C

1− CTnand Θn ≤ C ⇔ Vn ≤ Tn − C

CTn.

For example, in Figure 4.1 we depicted the curves v = C1−Ct and v = |t|−C

Ctfor C = 0.4. It at once follows from Figure 4.1 that in case Tn > 0 one hasthat min (Θn−1,Θn) < 0.4 < 1/

√5. Consequently, we only need to investigate

the case Tn < 0. In view of Borel’s result it is interesting to depict the curvesv = C

1−Ct and v = −t−CCt for C = 1/

√5, see also Figure 4.2.

− 12 0

g2

−g2

− 1√5

.................................................................................................................

.....................................................................................................

............................................................................................

....................................................................................

...........................................

..........................................................................................................................................................................................................................................................................................................

Figure 4.2: The curves v = 1√5−t

and v = − t√

5+1t .

Setting D = (x, y) ∈ Ω : x < 0, y > 1/(√

5− x), it follows from Figure 4.2that

(Tn, Vn) ∈ D ⇔ min(Θn−1, Θn) >1√5.

We have to consider the following two cases:

(i) − 12 ≤ Tn < − 2

5 . In this case bn+1 = 2, en+1 = −1 and en+2 = 1. From(4.7) we find that

Θn+1 =(1 + 2Tn)(2− Vn)

1 + TnVn.

Since ∂Θn+1/∂Tn = 0 if and only if Vn = bn+1, it follows that on

∆(−1, 1, 2) = [−12,−2

5)× [0, g2)

the approximation coefficient Θn+1 attains its maximum on the boundary of∆(−1, 1, 2) at (− 2

5 , 0), the maximum being 0.4 < 1/√

5. So on ∆(−1, 1, 2) ∩ Dwe see that min(Θn−1, Θn, Θn+1) < 1/

√5.

(ii) − 25 ≤ Tn < −g2. In this case bn+1 = 3, en+1 = −1, en+2 = −1, and

from (4.7) we find that

Θn+1 =(1 + 3Tn)(Vn − 3)

1 + TnVn.

Tong’s pre-spectrum for the NICF 67

Setting ∆(−1,−1, 3) = [− 25 ,− 1

3 )× [0, g2), we clearly are interested where Θn−1,Θn and Θn+1 attain their maximum on H = ∆(−1,−1, 3) ∩ D, and what thesmallest value of these three maxima is. It is easily verified that Θn−1,Θn andΘn+1 all attain their maximum on the boundary of H = ∆(−1,−1, 3) ∩ D in(−0.4,−g2). For Θn−1 the maximal value is 5g2/(5− 2g2) = 0.4508 · · · , for Θn

the maximal value is 2/(5− 2g2) = 0.4721 · · · , while for Θn+1 the maximum isg.

Since −g2 is a fixed-point under S, one sees that the NICF-expansion of −g2

is given by

−g2 = [ 0; −1/3, −1/3, · · · , −1/3, · · · ] = [ 0; −1/3 ],

where the bar indicates the period. But then we see that

− 25 ,− 5

13 = S−1(− 2

5

),− 13

34 = S−2(− 2

5

), . . . , S−n

(− 25

), . . . (4.8)

which are the NICF-convergents of −g2. Defining the Fibonacci numbers Fn,n ≥ 0, by F0 = 1, F1 = 1 and Fn+1 = Fn + Fn−1, n ≥ 1, one has that

3Fn−Fn−2 = Fn+2, n ≥ 2 and Fn =1√5

(Gn+1 − (−g)n+1

), n ≥ 0.

But then the sequence of convergents of −g2 from (4.8) is given by

tk = − F2k

F2k+2, k ≥ 1. (4.9)

Define for k ≥ 1Hk =

([tk,−g2)× [0, g2]

) ∩ D,

see also Figure 4.3.

.....................................................

..............................

............................................− 1

2 − 25 −g2− 5

13

g2

0.36

0.39

..........................................................................................................................................

..........................................

................................

............................

.................................

......................................

...............................................

......................................................................

.........................................................................................

..........................................

...........

Figure 4.3: The sets Hk for k ≥ 1.

68 TONG’S SPECTRUM FOR SRCF EXPANSIONS

For (Tn, Vn) ∈ H one has that

Tn =−1

3 + Tn+1and Vn = bn+1 − 1

Vn+1.

But then it follows that the curve y = 1/(√

5− x) in H is mapped by S to thecurve

y =3√

5 + 1 +√

5x

9√

5 + (3√

5− 1)x, − 1

2 ≤ x ≤ −g2,

in D, see also Figure 4.3. But from this it follows that

(Tn, Vn) ∈ Hk ⇔ min (Θn−1, Θn, . . . , Θn+k) >1√5.

For (Tn, Vn) ∈ Hk one has that Θn−1, Θn and Θn+1 all attain their maximumvalue on the boundary of Hk at (tk, g2), the minimum of these maxima being

τk =g2

1 + tk · g2=

g2F2k+2

F2k+2 − g2F2k, k ≥ 1.

Since G = 1/g and Fn = 1√5

(Gn+1 − (−g)n+1

)for n ≥ 0, we find that

τk( 12 ) =

g2(G2k+3 − (−g)2k+3

)

(G2k+3 − (−g)2k+3)− g2 (G2k+1 − (−g)2k+1)

=G2 + g4k+4

G4 − 1

=1√5

(1 + g4k+6

), k ≥ 1.

If (Tn, Vn) ∈ Hk \ Tk+1 is arbitrarily close to (tk, g2) for some k ≥ 2, thenS(Tn, Vn)) = (Tn+1, Vn+1) will be in Hk−1 \ Hk, and (Tn+1, Vn+1) will be arbi-trarily close to (tk−1, g

2). Continuing in this way we find that (Tn+2, Vn+2) ∈Hk−2 \ Tk−1 will be arbitrarily close to (tk−2, g

2), etc. Since τi < τi−1 for i ≥ 2,we see that for such a point (Tn, Vn) ∈ Hk \Hk+1 one has that Θn−1 < τk, whilemin(Θn, Θn+1, . . . , Θn+k) > τk( 1

2 ). This proves Tong’s Theorem 4.1.

As a corollary of our approach we have the following result, which is adirect consequence of the fact that for almost all x, the sequence (Tn, Vn)n≥1 isdistributed over Ω according to the density function d(x, y).

Theorem 4.2 For almost all x the limit

limn→∞

1n

#

1 ≤ j ≤ n : min(Θj−1, Θj , . . . , Θj+k) >1√5

exists, and equals

µ(Hk) =1

log G

∫ −g2

tk

(∫ g2

1√5−x

dy

(1 + xy)2

)dx

=1

log G

(log

1− g4

1 + g2tk+

tk + g2

√5

).

Tong’s spectrum for Nakada’s α-expansions 69

For example, for almost all x the limit

limn→∞

1n

#

1 ≤ j ≤ n : min(Θj−1, Θj , Θj+1) >1√5

exists, and equals µ(H) = 6.795 · · · × 10−5.

4.3 Tong’s spectrum for Nakada’s α-expansions

In 1981 Nakada generalized the RCF and NICF to a class of semi-regular con-tinued fraction expansions, the so-called α-expansions. Let 1

2 ≤ α ≤ 1, anddefine the map Sα : [α− 1, α) → [α− 1, α) by

Sα(x) =e1

x− d1, Sα(0) = 0,

where e1 = e1(x) = sgn(x), d1 = d1(x) = b|1/x| + 1 − αc. Setting dn =d1(Sn−1

α (x)) and en = e1(Sn−1α (x)), we see that repeated application of the map

Sα yields a semi-regular continued fraction expansion of any x ∈ [α−1, α)\0:the α-expansion of x. Note that α = 1/2 is the case of the NICF, while α = 1is the RCF case.

Setting

Ωα =

[α− 1,−g2)× [0, g2) ∪ [−g2, 1−2αα ]× [0, g2]

∪(1−2αα , 0)× [0, 1

2 ] if 12 ≤ α ≤ g,

∪[0, 2α−11−α ]× [0, 1

2 ) ∪ (2α−11−α , α)× [0, g)

[α− 1, α)× [0, 12 ] ∪ [0, 1−α

α ]× [0, 12 )

∪(1−αα )× [0, 1], if g < α ≤ 1,

we define a map Sα : Ωα → Ωα by

Sα(x, y) =(

Sα(x),1

d1 + e1y

), (x, y) ∈ Ωα.

Nakada showed in [Na] that(Ωα,Bα, µα, Sα

)forms an ergodic system,

where Bα is the collection of Borel sets of Ωα, and µα is an Sα-invariant proba-bility measure on (Ωα,Bα) with density dα(x, y) = 0 for (x, y) outside Ωα, andfor (x, y) ∈ Ωα

dα(x, y) =

1log G

1(1 + xy)2

, if 12 ≤ α ≤ g,

1log(1 + α)

1(1 + xy)2

, if g < α ≤ 1.

70 TONG’S SPECTRUM FOR SRCF EXPANSIONS

Let x be an irrational number, with α-expansion

[d0; e1/d1, e2/d2, · · · , en/dn, · · · ],where d0 ∈ Z is such that x− d0 ∈ [α− 1, α), and with α-convergents rn/sn =[d0; e1/d − 1, e2/d2, . . . , en/dn], for n ≥ 0. As before, the α-approximationcoefficients Θα,n(x) = Θn are defined by Θn = sn |snx− rn|, n ≥ 0. For x− d0

we define the sequences (Tn)n≥0 and (Vn)n≥0 by

(Tn, Vn) = Snα(x− d0, 0), n ≥ 0.

It is not difficult to show, see e.g. [K2], that (4.6) and (4.7) hold for any αbetween 1/2 and 1.

To find a Tong spectrum we consider the two cases 12 ≤ α ≤ g and g < α ≤ 1

separately. We start with the second—easier—case.

4.3.1 The case g < α ≤ 1

For these values the Tong spectrum is either finite, or void. To be more specific,we have the following theorem.

Theorem 4.3 Let g < α ≤ 1. Then for any irrational number x, with α-expansion x = [ d0; e1/d1, e2/d2, · · · , en/dn, · · · ].

(i) For any positive integer n for which Tn ≥ 0 one has

min (Θn−1,Θn, Θn+1) <1√5.

(ii) Let g < α < 10√

5−115√

5+1= 0.618414858 · · · . and define for ` ≥ 0

(x`, y`) = S`α

(α− 1, 1

2

)and ϑ`(α) =

(1 + 3x`)(y` − 3)1 + x`y`

.

Let m be the smallest non-negative integer for which ϑm+1(α) ≤ 1/√

5 < ϑm(α).Then

min(Θn−1, Θn, Θn+1, . . . , Θn+j+1) < ϑj(α) for j = 0, 1, . . . ,m.

(iii) Let 10√

5−115√

5+1≤ α ≤ 1. For any positive integer n one has

min (Θn−1,Θn, Θn+1) <1√5.

Proof.(i) Let

D+ =

(x, y) ∈ Ωα;x ≥ 0,x

1 + xy>

1√5,

y

1 + xy>

1√5

,

Tong’s spectrum for Nakada’s α-expansions 71

............................................... ................

.................................................

................

α− 1 0− 1√5

1√5

1−αα

α

12

D+

D−

g

1

...................................................................................................

.......................................................................................

............................................................................

....................................................................

.............................................................

.....................................................

..................................................

..............................................

..............................

...................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................

Figure 4.4: The natural extension of the α-expansion for α = 0.63, and thecurves v = 1√

5−tand v = |t|√5−1

t .

see also Figure 4.4. Then by definition of D+

(Tn, Vn) ∈ D+ ⇔ min (Θn−1,Θn) >1√5.

On D+ one has that en+1 = 1 = en+2, dn+1 = 1, and therefore

Θn+1 =(1− Tn)(1 + Vn)

1 + TnVn.

One easily sees that on D+ the approximation coefficient Θn+1 attains its max-imum on the boundary of D+ in (g, g), the maximum being 1/

√5. This proves

(i).

In case g < α ≤ (21− 4√

5)/19 = 0.634512004 · · · , define D− by

D− =

(x, y) ∈ Ωα;x ≤ 0,|x|

1 + xy>

1√5,

y

1 + xy>

1√5

,

see also Figure 4.4. One has

(Tn, Vn) ∈ D− ⇔ min (Θn−1, Θn) >1√5,

72 TONG’S SPECTRUM FOR SRCF EXPANSIONS

and (Tn, Vn) ∈ D− implies that en+1 = −1 = en+2, dn+1 = 3. We find that

Θn+1 =(1 + 3Tn)(Vn − 3)

1 + TnVn. (4.10)

(ii) For g < α ≤ 10√

5−115√

5+1= 0.618414858 · · · the set

T − =

(x, y) ∈ Ωα ; α− 1 ≤ x ≤ 0,

9√

5x + 3√

5 + 1(3√

5− 1)x +√

5< y ≤ 1

2

is a non-empty set contained in D−.One easily sees that on T − the approximation coefficients Θn−1, Θn and

Θn+1 are all larger than 1/√

5, and that they attain their maximum on theboundary ofD− at (α−1, 1

2 ). The smallest of these three maxima is ϑ0(α), whichis given by Θn+1. Note that for 1 ≤ j ≤ m one has that xj−1 = Sj−1(α− 1) <Sj(α − 1) = xj , and that y0 = 1

2 , y1 = 513 , y3 = 13

34 , . . . , ym are the first m + 1RCF-convergents of g2 which are larger than g2. From (4.10) it follows that for(Tn, Vn) ∈ D− one has that Θn+1 = 1/

√5 if and only if (Tn, Vn) is on the curve

y =9√

5x + 3√

5 + 1(3√

5− 1)x +√

5, α− 1 ≤ x ≤ 2−4

√5

19 .

This curve enters T − in (α − 1, 9√

5(α−1)+3√

5+1

(3√

5−1)(α−1)+√

5)), and since the second coor-

dinate of this point is larger that g2, we see that a smallest non-negative m forwhich ϑm+1(α) ≤ 1/

√5 < ϑm(α) exists. If Sα(D−) ∩ T − 6= ∅, we find that

min(Θn−1, Θn, Θn+1, Θn+2) < ϑ1(α), etc. This proves (ii).

(iii) In case 10√

5−115√

5+1< α ≤ 21−4

√5

19 the set T − is empty. Together with case

(i) we find that min(Θn−1,Θn, Θn+1) < 1/√

5.In case (21− 4

√5)/19 < α ≤ 1 the set D− is empty. In this case one has for

Tn < 0 that

Θn <2− 2α

1 + α<

1√5.

which—together with (i)—implies (iii). 2

As a corollary of our method we have the following result.

Proposition 4.1 Let g ≤ α ≤ 10√

5−115√

5+1. For almost all x the limit

limn→∞

1n

#

1 ≤ j ≤ n : min(Θj−1, Θj , Θj+1) >1√5

Tong’s spectrum for Nakada’s α-expansions 73

exists, and equals

µα(T −) =1

log G

∫ − 5√

5+215√

5+1

α−1

(∫ 12

9√

5x+3√

5+1(3√

5−1)x+√

5

dy

(1 + xy)2

)dx

=1

log G

(log

(5√

52− 3α

α + 1

)+

13√

5

(1

2− 3α− 15

√5 + 15

)).

Remark 4.1 Note that for α = g we find that µα(T −) = 6.795 · · · × 10−5,which is the same value as for the NICF, see also page 69. For the α’s underconsideration, this is the maximal possible value for µα(T −).

4.3.2 The case 12≤ α ≤ g

In this case there is a Tong spectrum for every α-expansion. To be more specific,we have the following theorem.

Theorem 4.4 Let 12 ≤ α ≤ g. Then for any irrational number x, with α-

expansion x = [ d0; e1/d1, e2/d2, · · · , en/dn, · · · ], we have the following.

(i) For any positive integer n for which Tn ≥ 0 one has

min (Θn−1, Θn) <1√5.

(ii) Let 12 ≤ α < −9−26g+

√1092g+2457

50 = 0.617888713 · · · . Then for all posi-tive integers n and k one has

min(Θn−1, Θn, . . . , Θn+k) <1√5

(1 +

g4k(g − α)G + α

).

The constant τk(α) = 1√5

(1 + g4k(g−α)

G+α

)is best possible.

(iii) Let −9−26g+√

1092g+245750 < α < g, and define for ` ≥ 1

(x`, y`) = S`−1α

(1−2α

α , 12

)and ϑ`(α) =

(1 + 3x`)(y` − 3)1 + x`y`

.

Let m be the smallest positive integer for which ϑm+1(α) ≤ 1/√

5 < ϑm(α), andset

%k(α) =

max(ϑk(α), τk(α)), if k = 1, . . . ,m,τk(α), if k ≥ m + 1.

Then

min(Θn−1, Θn,Θn+1, . . . , Θn+j+1) < %k(α) for k ≥ 1.

74 TONG’S SPECTRUM FOR SRCF EXPANSIONS

(iv) For α = g the Tong spectrum is identical to the Tong spectrum forα = 1

2 . To be precise, for all positive integers n and k one has

min(Θn−1, Θn, . . . , Θn+k) <1√5

+1√5

(3−√5

2

)2k+3

.

The constant τk(g) = 1√5

+ 1√5

(3−√5

2

)2k+3

is best possible.

Proof. The proof of Theorem 4.4 bears strong resemblance to that of Theorem4.3; it is in some cases more complicated. We therefore restrict ourselves tooutlining the proof.

(i) For x > 0 one has that the lines

y =1√

5− xand y =

√5x− 1

x

intersect at (g, g), which is outside Ωα for every α ∈ [ 12 , g], see also Figure 4.5.Consequently, for every positive integer n for which Tn ≥ 0 one has that

min (Θn−1,Θn) <1√5.

..................................................... ................

α− 1 0

12

g

g2

α1−2αα

2α−11−α

D`

...........................................................................................................

...........................................................................................

................................................................................

......................................................................

................................................................

.......................................................

....................................................

.................................................

..........................................

.....

...................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................

Figure 4.5: The natural extension of the α-expansion for α = 0.55, and thecurves v = 1√

5−tand v = |t|√5−1

t .

(ii) Similarly as in the proofs of Theorems 4.1 and 4.3 we define regions D`,Du and T by

D` =

(x, y) ∈ Ωα;α− 1 ≤ x ≤ −g2,|x|

1 + xy>

1√5,

y

1 + xy>

1√5

,

Du =

(x, y) ∈ Ωα;−g2 ≤ x ≤ 0,|x|

1 + xy>

1√5,

y

1 + xy>

1√5

,

Tong’s spectrum for Nakada’s α-expansions 75

and

T =

(x, y) ∈ Ωα ;

1− 2α

α≤ x ≤ 0,

9√

5x + 3√

5 + 1(3√

5− 1)x +√

5< y ≤ 1

2

,

see also Figure 4.5 where D` is depicted (both Du and T are empty in case α =0.55). Note that Du is an empty set for α ∈ [ 12 , 2

√5+1

4√

5) = [0.5, 0.611803398 · · · ),

and that T is empty for α ∈ [ 12 , 15√

5+125√

5) = [0.5, 0.617888543 · · · ). The role

of D` in this proof is comparable to that of the set D in the proof of Tong’sTheorem 4.1 in Section 4.2, the roles of Du and T are comparable to those ofD− respectively T − in the proof of Theorem 4.3. As in the proof of Theorem4.1 the set Du is not really important; what matters is its subset T .

We first consider for every α ∈ [ 12 , g) the set D` (for α = g this set consistsof only one point: (−g2, g2)). Setting

∆(−1, 1, 2) = [α− 1,−1

α + 2)× [0, g2),

we have that dn+1 = 2, en+1 = −1 and en+2 = 1 for (Tn, Vn) ∈ ∆(−1,−1, 2).Similar to the proof of Theorem 4.1 we find that Θn+1 attains its maximum onD`∩∆(−1,−1, 2) on the boundary of this set at ( −1

α+2 , α+2(α+2)

√5+1

), the maximum

being α((2√

5−1)α+4√

5)√5(α+2)2

, which is smaller than 1/√

5 for α ∈ [ 12 , g) (and equals

1/√

5 for α = g).Setting ∆(−1,−1, 3) = [− 1

α+2 ,− 13 )× [0, g2] and H = ∆(−1,−1, 3)∩D`, one

finds that Θn−1, Θn and Θn+1 attain on H their maxima on the boundary ofH at (− 1

α+2 , g2). These maxima are (α+2)g2

α+G , 1α+G , respectively (1−α)G2

α+G . Thesmallest of these three maxima is given by Θn−1, and equals

(α + 2)g2

α + G>

1√5, for 1

2 ≤ α < g.

Note that equality holds for α = g.Similar to the proof of Theorem 4.1 we have that the sequence

t1 =−1

α + 2, t2 = S−1

α (t1) =−(α + 2)3α + 5

, t3 = S−2α (t1) =

−(3α + 5)8α + 13

, . . .

converges from below to −g2. Setting F−1 = 0 we find

tk = S−(k−1)α (t1) =

−(F2k−3α + F2k−2)F2k−1α + F2k

, k ≥ 1.

Again, define Hk by

Hk =([tk,−g2)× [0, g2]

) ∩ D`, k ≥ 1.

76 TONG’S SPECTRUM FOR SRCF EXPANSIONS

Then it follows that

(Tn, Vn) ∈ Hk ⇔ min (Θn−1, Θn, . . . , Θn+k) >1√5.

For (Tn, Vn) ∈ Hk one has that Θn−1, Θn and Θn+1 all attain their maximumvalue on the boundary of Hk at (tk, g2), the minimum of these maxima being

τk(α) =g2

1 + tk · g2=

g2 (F2k−1α + F2k)F2k−1α + F2k − (F2k−3α + F2k−2)

=1√5

(1 +

g4k(g − α))G + α

), k ≥ 1.

By definition T is that part of Ωα, which is enclosed by the lines x = 1−2αα

and y = 12 , and the curve y = 9

√5x+3

√5+1

(3√

5−1)x+√

5. But then we see that T is empty

for α ∈ [ 12 , 15√

5+125√

5) = [0.5, 0.617888543 · · · ). Therefore, for these values of α

we find for all positive integers n and k that min(Θn−1,Θn, . . . , Θn+k) < τk(α),and that the constant τk(α) cannot be replaced by a smaller constant.

In case T is non-empty, we have for (Tn, Vn) ∈ T that

min(Θn−1, Θn,Θn+1) ≤ 25α− 15.

Recall that for (Tn, Vn) ∈ D` one has that

min(Θn−1,Θn, Θn+1) ≤ (α + 2)g2

α + G,

and therefore for α ∈ [ 15√

5+125√

5, −9−26g+

√1092g+2457

50 ] it follows that ϑ1(α) ≤τ1(α), and that

(1− 2α

α,12

)=

(5α− 31− 2α

,25

)/∈ T ,

i.e., ϑ2(α) < 1/√

5 < τ2(α). Thus for all positive integers n and k we see that

min(Θn−1,Θn, . . . , Θn+k) < τk(α),

and that the τk(α) cannot be replaced by a smaller constant. This proves (ii).

(iii) For α ∈ [−9−26g+√

1092g+245750 , g) it follows from the definition of m and

the proof of (ii) that min(Θn−1,Θn, . . . , Θn+k) < τk(α) for k ≥ m + 1. Clearly,if (Tn, Vn) /∈ D` ∪ Du, then min(Θn−1,Θn) < 1/

√5, and we also have that

(Tn, Vn) /∈ D`∪T implies that min(Θn−1,Θn, Θn+1) < 1/√

5. For k = 1, 2 . . . , mwe moreover have that

(Tn, Vn) ∈ D` ⇒ min(Θn−1,Θn, . . . , Θn+k) < τk(α)

and(Tn, Vn) ∈ T ⇒ min(Θn−1, Θn, . . . , Θn+k) < ϑk(α).

Semi-regular continued fractions 77

This proves (iii).

(iv) The case α = g is an immediate consequence of the fact that

Skg

(−g2, 12

)= (−g2,−tk), for k ≥ 1,

where the rationals tk are given by (4.9). 2

As a corollary of our method we have the following result.

Proposition 4.2 Let 12 ≤ α ≤ g. For almost all x the limit

limn→∞

1n

#

1 ≤ j ≤ n : min(Θj−1, Θj , Θj+1) >1√5

exists, and equals

µα(D` ∪ T ) =

µα(D`), if 12 ≤ α < 15

√5+1

25√

5,

µα(D`) + µα(T ), if 15√

5+125√

5≤ α ≤ g,

where

µα(D`) =1

log G

∫ −g2

−1α+2

(∫ g2

1√5−x

dy

(1 + xy)2

)dx

=1

log G

(log

(1− g4

1− g2

α+2

)+

g2 − 1α+2√5

),

and

µα(T ) =1

log G

∫ − 5√

5+215√

5+1

1−2αα

(∫ 12

9√

5x+3√

5+1(3√

5−1)x+√

5

dy

(1 + xy)2

)dx

=1

log G

(log

(5√

5(5α− 3))

+1

3√

5

5α− 3− 15

√5 + 15

)).

Remark 4.2 Note that for α = 12 (the case of the NICF) and α = g we find

that µα(T ) = 6.795 · · · × 10−5, see also Remark 4.1 and page 69.

4.4 Semi-regular continued fractions

It can be shown, see e.g. [DK], that every semi-regular continued fraction ex-pansion can be obtained via two processes, the singularization process, and theinsertion process. For instance, in [K2] it is shown that the nearest integer

78 TONG’S SPECTRUM FOR SRCF EXPANSIONS

continued fraction expansion (and any other α-expansion for α ∈ [ 12 , 1]) can beobtained from the RCF via a well-described singularization algorithm, in termsof a certain subset of the natural extension of the RCF. In [K2] it is shown, thatonly partial quotients equal to 1 can be ‘singularized’, and since the effect ofthe singularization of the (n + 1)th partial quotient is the removal of the nthconvergent, we see that only relatively large convergents are removed while ap-plying a singularization process. To some extend this explains why the nearestinteger continued fraction has a Tong spectrum; one has thrown out relatively‘good’ convergents, and kept relatively ‘bad’ ones. That this is indeed the caseis shown by Bosma’s optimal continued fraction expansion (OCF). In this con-tinued fraction one throws out exactly as many regular convergents as in thenearest integer continued fraction expansion, but one has that for all irrationalnumbers x and all n ≥ 1 the approximation coefficients Θn, n ≥ 0, of the OCFexpansion of x satisfy

min(Θn−1,Θn) <1√5,

see [BK] for more details.As soon as one has the natural extension of a semi-regular continued fraction

expansion, one can mimic the approach of the previous section to see whetherthis particular SRCF expansion has a Tong spectrum. In fact, this also appliesto continued fraction expansions related to other groups, such as the Rosenexpansions. One can use the natural extension from [BKS] to show that eachRosen expansion has its own Tong spectrum, see also [KNS].

Bibliography

[A] Adams, William W. — On a relationship between the convergents of thenearest integer and regular continued fractions, Math. Comp. 33 (1979),no. 148, 1321–1331. MR 82g:10078

[BJW] Bosma, W., Jager, H. and Wiedijk, F. — Some metrical observationson the approximation by continued fractions, Nederl. Akad. Wetensch.Indag. Math. 45 (1983), no. 3, 281–299. MR 85f:11059

[BK] Bosma, Wieb, Kraaikamp, Cor — Metrical theory for optimal continuedfractions, J. Number Theory 34 (1990), no. 3, 251–270. MR 91d:11095

[B] Burger, E.B. — Exploring the number jungle: a journey into Diophan-tine analysis, Student Mathematical Library, 8. American MathematicalSociety, Providence, RI, 2000. MR 2001h:11001

[BKS] Burton, R.M., Kraaikamp, C. and Schmidt, TA. — Natural extensionsfor the Rosen fractions, Trans. Amer. Math. Soc. 352 (2000), no. 3,1277–1298. MR 2000j:11123

[CF] Cusick, T.W. and Flahive, M.E. — The Markoff and Lagrange spectra,Mathematical Surveys and Monographs, 30. American MathematicalSociety, Providence, RI, 1989. MR 90i:11069

[DK] Dajani, Karma, Kraaikamp, Cor — “The mother of all continued frac-tions”, Colloq. Math. 84/85 (2000), , part 1, 109–123. MR 2001h:11100

[J] Jager, H. — On the speed of convergence of the nearest integer continuedfraction, Math. Comp. 39 (1982), no. 160, 555–558. MR 83k:10094

[JK] Jager, H. and Kraaikamp, C. — On the approximation by continuedfractions, Indag. Math. 51 (1989), no. 2, 289-307. MR 90k:11084

[KS] Kopetzky, H.G. and Schnitzer, F.J. — Eine geometrische Methode beider Approximation durch Kettenbruche nach dem nachsten Ganzen,Math. Pannon. 6 (1995), no. 1, 45–54. MR 97c:11077

[KNS] Kraaikamp, Cor, Nakada, Hitoshi, Schmidt, Tom — Tong spectra forRosen fractions, preprint 2002.

79

80 TONG’S SPECTRUM FOR SRCF EXPANSIONS

[K1] Kraaikamp, C. — The distribution of some sequences connected withthe nearest integer continued fraction, Indag. Math. 49 (1987), 177-191.MR 88j:11045

[K2] Kraaikamp, C. — A new class of continued fraction expansions, ActaArith. 57 (1991), no. 1, 1–39. MR 92a:11090

[Na] Nakada, H. — Metrical theory for a class of continued fraction trans-formations and their natural extensions, Tokyo J. Math. 4 (1981), no.2, 399–426. MR 83k:10095

[T1] Tong, Jing Cheng — Approximation by nearest integer continued frac-tions, Math. Scand. 71 (1992), no. 2, 161–166. MR 94c:11059

[T2] Tong, Jing Cheng — Approximation by nearest integer continued frac-tions (II), Math. Scand. 74 (1994), no. 1, 17–18. MR 95c:11085

Chapter 5

A Note on HurwitzianNumbers

5.1 Introduction

It is well-known that every real irrational number x has a unique regular con-tinued fraction expansion of the form

x = [ a0; a1, · · · , an, · · · ], (5.1)

where a0 ∈ Z is such that x− a0 ∈ [0, 1), and an ∈ N for n ≥ 1. The number xis called Hurwitzian if (5.1) can be written as

x = [ a0; a1, · · · , an, an+1(k), · · · , an+p(k) ]∞k=0, (5.2)

where an+1(k), . . . , an+p(k) (the so-called quasi period of x) are polynomialswith rational coefficients which take positive integral values for k = 0, 1, 2, . . .,and at least one of them is not constant. By the bar we mean that an+i+kp =an+i(k), where 1 ≤ i ≤ p and k ≥ 0. A well-known example of such numbers ise = [2; 1, 2k + 2, 1]∞k=0; see [P] for more examples. Hurwitzian numbers are gen-eralizations of numbers with eventually periodic continued fraction expansions.An old and classical result states that a number x is a quadratic irrational (thatis, an irrational root of a polynomial of degree 2 with integer coefficients) if andonly if x has a continued fraction expansion which is eventually periodic, i.e., ifx is of the form

x = [a0; a1, · · · , ap, ap+1, · · · , ap+`], p ≥ 0, ` ≥ 1, (5.3)

where the bar indicates the period, see [HW], [O] or [P] for various classicalproofs of this result.

Apart from the regular continued fraction (RCF) expansion of x there arevery many other—classical—continued fraction expansions of x, such as thenearest integer continued fraction (NICF) expansion, the ‘backward’ continued

81

82 A NOTE ON HURWITZIAN NUMBERS

fraction expansion, and Nakada’s α-expansions. In this note we will define whatHurwitzian numbers are for such continued fraction expansions and show thattheir set of Hurwitzian numbers coincides with the classical set of Hurwitziannumbers. As a by-product quadratic irrationals will have an eventually periodexpansion for each of these expansions.

5.2 Hurwitzian numbers for the NICF

Every x ∈ R \Q can be expanded in a unique continued fraction expansion

x = b0 +e1

b1 +e2

b2 +.. . +

en

bn +.. .

=: [ b0; e1/b1, e2/b2, · · · , en/bn, · · · ] ,

satisfying b0 ∈ Z, x − b0 ∈ [− 12 , 1

2 ), en = ±1, bn ∈ N and en+1 + bn ≥ 2for n ≥ 1. This continued fraction expansion is known as the nearest integercontinued fraction (NICF) expansion of x.

In [K] it is shown that the NICF expansion can be obtained from the RCFby singularizing the first, the third, etc. 1’s in every block of consecutive 1’spreceded by either a partial quotient different from 1 or preceded by a0. Thissingularization process is based upon the identity

A +e

1 +1

B + ξ

= A + e +− e

B + 1 + ξ.

Example 5.1 The NICF expansion of e is given by

[3;−1/4, −1/2, 1/(2k + 5)]∞k=0.

In view of this example we have the following definition.

Definition 5.1 Let x ∈ R \Q. Then x has an NICF-Hurwitzian expansion if

x = [ b0; e1/b1, · · · , en/bn, en+1/bn+1(k), · · · , en+p/bn+p(k) ]∞k=0

where b0 ∈ Z, x− b0 ∈ [− 12 , 1

2 ), en = ±1, bn ∈ N and en+1 + bn ≥ 2 for n ≥ 1.Moreover, for i = 1, . . . , p we have that bn+i(k) are polynomials with rationalcoefficients which take positive integral values for k = 0, 1, 2, . . ., and at leastone of them is non-constant.

The following result gives the necessary and sufficient condition for an irra-tional number to have an NICF-Hurwitzian expansion.

Theorem 5.1 Let x ∈ R \ Q. Then x is Hurwitzian if and only if x has anNICF-Hurwitzian expansion.

NICF-Hurwitzian expansions 83

Proof. Let x be a Hurwitzian number with RCF expansion given by (5.1) and(5.2). Let m0 ∈ N, m0 ≥ n, be such that am0 > 1. Note that (5.2) can bewritten as

x = [a0; a1, · · · , am0 , an+1(k), · · · , an+p(k)]∞k=0, (5.4)

where a1, . . . , am0 are positive integers, and where an+1(k), . . . , an+p(k) arepolynomials with rational coefficients which take positive integral values fork = 0, 1, 2, . . ., and at least one of them is not constant. Suppose moreover thatm0 is chosen in such a way, that for all k ≥ 0 all the non-constant polynomialsin the quasi-period an+1(k), . . . , an+p(k) have values greater than 1.

For i ∈ 1, . . . , p− 1, we consider two cases:Case (i): am0+i = 1. By definition of a Hurwitzian number there exist numbersj1 ∈ 0, 1, . . . , i− 1 and j2 ∈ i + 1, . . . , p for which am0+j1 > 1, am0+j2 > 1,and

am0+j1+1 = · · · = am0+i = · · · = am0+j2−1 = 1. (5.5)

In case i− j1 is odd the digit am0+i = 1 will be singularized, and in case i− j1 iseven it will not be singularized, but it will either change into −1/2 if j2 = i+1,or into −1/3 if j2 ≥ i + 2. Due to the quasi-periodicity and by definition of m0

we have for each k ∈ N that

am0+j1+kp+1 = · · · = am0+i+kp = · · · = am0+j2+kp−1 = 1,

and each of these blocks is singularized in the same way as the block (5.5) wassingularized, which means the same thing will happen to am0+i+(k−1)p = 1 forall k ∈ N.

Case (ii): am0+i > 1 (am0+i is either a constant or a polynomial). We have fourpossible cases:

(a) am0+i−1 = 1 = am0+i+1. In this case, am0+i−1 = 1 belongs to a block of1’s and will be singularized if and only if this block has odd length. Onthe other hand, am0+i+1 = 1 will always be singularized, so that am0+i

will either become −1/(am0+i + 2) (if the block of 1’s ‘before’ am0+i hasodd length), or becomes 1/(am0+i + 1).

(b) am0+i−1 6= 1 = am0+i+1. In this case, am0+i becomes 1/(am0+i + 1), dueto the singularization of am0+i+1 = 1.

(c) am0+i−1 = 1 6= am0+i+1. In this case, am0+i becomes either −1/(am0+i +1), or remains unchanged, depending on whether am0+i−1 = 1 is singular-ized or not.

(d) am0+i−1 6= 1 6= am0+i+1. In this case am0+i will remain unchanged.

Due to the periodicity the same thing will happen to am0+i+(k−1)p > 1 for allk ∈ N.

To conclude, from (i) and (ii) we see that for each i ∈ 1, . . . , p and for allk ∈ N one has exactly one of the following possibilities:

84 A NOTE ON HURWITZIAN NUMBERS

— am0+i+(k−1)p = 1 always disappears due to a singularization;

— am0+i+(k−1)p > 1 always remains unchanged;

— am0+i+(k−1)p > 1 always becomes −1/(am0+i+(k−1)p + 1) due to the sin-gularization of a digit 1 before it;

— am0+i+(k−1)p = 1 always becomes 1/(am0+i+(k−1)p + 1) due to the singu-larization of a digit 1 after it;

— am0+i+(k−1)p = 1 always becomes −1/(am0+i+(k−1)p + 2) due to the sin-gularization of a digit 1 before and after it.

Thus we obtain a quasi-period for the NICF expansion of x.Conversely, since the singularization process can be reversed in a unique way,

we see that a NICF-Hurwitzian number x is also Hurwitzian. 2

Applying the procedure given in the proof of Theorem 1 yields that the NICF-expansion of e is given by e = [3;−1/4, −1/2, 1/(2k + 5),−1/2]∞k=0, which isanother way of writing e in Example 5.1.

From the proof of Theorem 5.1 it is at once clear that x is a quadraticirrational if and only if the NICF-expansion of x is eventually periodic.

5.3 Hurwitzian numbers for the backward con-tinued fraction

Every x ∈ R \Q can be expanded in a unique continued fraction expansion

c0 −1

c1 −1

c2 −. . . − 1

cn −. . .

=: [ c0; −1/c1, −1/c2, · · · , −1/cn, · · · ] ,

where c0 ∈ Z such that x− c0 ∈ [−1, 0) and ci’s are all integers greater than 1.This continued fraction is known as the backward continued fraction expansionof x; see [DK] for details.

Proposition 2 in [DK] gives an algorithm yielding the backward continuedfraction expansion from the regular one using singularizations and insertions.The latter is based on the following identity.

A +1

B + ξ= A + 1 +

− 1

1 +1

B − 1 + ξ

.

From this algorithm it follows that x = [a0; a1, a2, · · · ] has as backward expan-sion

[a0 + 1; (−1/2)a1−1, −1/(a2 + 2), (−1/2)a3−1, −1/(a4 + 2), · · · ] (5.6)

Backward-Hurwitzian expansions 85

where (−1/2)t is an abbreviation of −1/2, · · · ,−1/2︸ ︷︷ ︸t−times

for t ≥ 1. In case t = 0,

the term (1/2)t should be omitted.

Example 5.2 The backward expansion of e is given by

[3; −1/(4k + 4), −1/3, (−1/2)4k+3, −1/3 ]∞k=0 .

This example leads to the following definition.

Definition 5.2 Let x ∈ R\Q. Then x has a backward-Hurwitzian expansion if

x = [ c0; (−1/c1)r1 , · · · , (−1/cn)rn ,

(−1/cn+1(k))rn+1(k), · · · , (−1/cn+p(k))rn+p(k) ]∞k=0

where c0 ∈ Z such that x−c0 ∈ [−1, 0); (ci, ri) = (c, 1) or (2, r) for i = 1, . . . , n,where c is an integer greater than 2 and r a positive integer. We call p the‘length’ of the quasi-period. Moreover,

(cn+i(k), rn+i(k)) = (fi(k), 1) or (2, gi(k))

for i = 1, . . . , p where fi(k) and gi(k) are polynomials with rational coefficientswhich take positive integral values for k = 0, 1, 2, . . . and at least one of them isnot constant. Here (−1/c)r is an abbreviation of −1/c, · · · , −1/c︸ ︷︷ ︸

r−times

.

The following result gives the necessary and sufficient condition for an irra-tional number to have a backward-Hurwitzian expansion.

Theorem 5.2 Let x ∈ R \ Q. Then x is Hurwitzian if and only if x has abackward-Hurwitzian expansion.

Proof. Let x be a Hurwitzian number, with RCF-expansion (5.1). We firstnote that (5.6) yields that an in the RCF-expansion of x becomes (−1/2)an−1

in the backward expansion of x if n is odd, and becomes −1/(an + 2) if n iseven. Let m0 be defined as in the proof of Theorem 5.1. Then for all i > m0

we observe the following:

(i) If ai = 1, then it either disappears in case i is odd, or becomes −1/3 incase i is even.

(ii) If ai > 1, then it either becomes (−1/2)ai−1 in case i is odd, or −1/(ai+2)in case i is even.

Let p be the length of the quasi-period of the RCF-expansion of x. We see thatfor all k ∈ N the same thing will happen to each ai+(k−1)p if p is even or toeach ai+2(k−1)p if p is odd, which yields a quasi-periodicity for the backwardexpansion of x.

86 A NOTE ON HURWITZIAN NUMBERS

Conversely, since the singularization and insertion processes can be reversedin a unique way, we see that a backward-Hurwitzian number x is also Hur-witzian. 2

Clearly x is a quadratic irrational if and only if the backward-expansion ofx is eventually periodic.

The next section gives a generalization of Section 5.2.

5.4 Hurwitzian numbers for α-expansions

In this section we will define Hurwitzian numbers for the so-called α-expansions,of which the nearest integer continued fraction expansion is an example. Theseα-expansions were introduced and studied by H. Nakada in 1981 ([N]). We willshow that Hurwitzian numbers for these α-expansions also coincide with theclassical Hurwitzian numbers.

For α ∈ [1/2, 1], let x ∈ [α− 1, α] and define

f1 = f1(x) := b|1/x|+ 1− αc, x 6= 0,

fn = fn(x) := f1(Tn−1α (x)), n ≥ 2, Tn−1

α (x) 6= 0,(5.7)

where Tα : [α− 1, α] → [α− 1, α] is defined by

Tα = |1/x| − b|1/x|+ 1− αc

and bξc denotes the largest integer not exceeding ξ.Every x ∈ R \Q can be expanded in a continued fraction expansion

x = [ f0; e1/f1, e2/f2, · · · , en/fn, · · · ],

where f0 ∈ Z, x − f0 ∈ [α − 1, α), en = ±1, fn ∈ N, n ≥ 1, are given by (5.7).We call this continued fraction α-expansion of x.

Remark 5.1 Note that for α = 1/2 one has the NICF-expansion, while α = 1is the RCF case.

In [K] it is shown that α-expansions can be viewed as S-expansions, withsingularization areas

Sα = [α, 1]× [0, 1], if g < α ≤ 1

and

Sα = [α, g)× [0, g) ∪ [g, (1− α)/α]× [0, g] ∪ ((1− α)/α, 1]× [0, 1]

in case 1/2 ≤ α ≤ g, where g = (√

5− 1)/2; see Figure 5.1.

α-Hurwitzian expansions 87

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

12

g α0

1

0 1

1

g

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

12

gα1 1−αα

.......................................................................................................................

(i) g < α ≤ 1 (ii) 1/2 ≤ α ≤ g

Figure 5.1: Singularization areas for α-expansions

In general a singularization area S is a subset of the so-called natural ex-tension [0, 1) × [0, 1] of the RCF-expansion, which satisfies the following threeconditions:

(i): S ⊂ [ 12 , 1)× [0, 1]; (ii): T (S) ∩ S = ∅ and (iii): λ(∂S) = 0.Here λ is Lebesgue measure on [0, 1)× [0, 1], and T : [0, 1)× [0, 1] → [0, 1)× [0, 1]is the natural extension map of the RCF-expansion, given by

T (x, y) =(

1x− b 1

xc, 1b 1

xc+ y

), (x, y) ∈ (0, 1)×[0, 1]; T (0, y) = (0, 0), y ∈ [0, 1].

Let x ∈ [0, 1), with RCF-expansion [a0; a1, a2, · · · ]. Then the S-expansion of xis obtained via the following algorithm:

singularize an+1 = 1 if and only if (Tn, Vn) ∈ Sα,

where Tn = [0; an+1, an+2, · · · ] and Vn = [0; an, · · · , a1], i.e., (Tn, Vn) = T n(x, 0),for more details, see [K].

The following lemma is very handy.

Lemma 5.1 Let x, y ∈ [0, 1), with RCF-expansions

x = [0; a1(x), a2(x), · · · ], y = [0; a1(y), a2(y), · · · ].

Let x 6= y and k ∈ N ∪ 0 be such that

a1(x) = a1(y), . . . , ak−1(x) = ak−1(y), and ak(x) 6= ak(y).

88 A NOTE ON HURWITZIAN NUMBERS

Then one has

x > y if and only if

ak(x) < ak(y) if k is odd,

ak(x) > ak(y) if k is even.

Proof. For n ∈ N, a1, . . . , an ∈ N, define cylinders ∆n(a1, . . . , an) by

∆n(a1, . . . , an) = x ∈ [0, 1) ; a1(x) = a1, . . . , an(x) = an.

For x, y ∈ ∆k−1(a1, . . . , ak−1), x < y, one has by definition of the RCF-mapT = T1 that T (x), T (y) ∈ ∆k−2(a2, . . . , ak−1), and T (x) > T (y). Repeatingthis argument k−2-times, we find that T k−2(x), T k−2(y) ∈ ∆1(ak−1), and thatT k−2(x) < T k−2(y) if and only if k is even. Since T (∆1(ak−1)) = [0, 1) andak(x) 6= ak(y), it follows from the definition of T that T k−1(x) > T k−1(y) ifand only if k is even. Since T k−1(x) ∈ ∆1(ak(x)) =

(1

ak+1 , 1ak

], and T k−1(y) ∈

∆1(ak(y)), it follows that ak(x) < ak(y) if and only if k is even. 2

We now define Hurwitzian numbers for α-expansions.

Definition 5.3 Let x ∈ R \ Q. Then, for a fixed α ∈ [1/2, 1], x has an α-Hurwitzian expansion if

x = [ f0; e1/f1, · · · , en/fn, en+1/fn+1(k), · · · , en+p/fn+p(k) ]∞k=0 (5.8)

is the α-expansion of x, where f0 ∈ Z, x − f0 ∈ [α − 1, α), en = ±1, fn ∈ N,n ≥ 1, are given by (5.7). Moreover, for i = 1, . . . , p we have that fn+i arepolynomials with rational coefficients which take positive integral values for k =0, 1, 2, . . ., and at least one of them is non-constant.

We have the following theorem.

Theorem 5.3 Let x ∈ R \ Q. Then x is Hurwitzian if and only if x has anα-Hurwitzian expansion.

Proof. As in the proof of Theorem 5.1, let m0 ∈ N be such that am0 >1, and for all m ≥ m0 all the non-constant polynomials in the quasi-periodan+1(k), . . . , an+p(k) of the RCF-expansion (5.4) of x have values greater than1. Let k ∈ m0 + 1, . . . , m0 + p be such that ak = 1. Then

T k−1(x) = [0; 1, ak+1, · · · ].

Case 1: g < α ≤ 1. In this case ak = 1 must be singularized if and only ifT k−1(x) ≥ α.

Clearly there exists a minimal i ∈ 1, . . . , p such that ak+i is a value of anon-constant polynomial. Further, let j ∈ N∪ ∞ be the first index such that

ak+j 6= aj+1(α)

α-Hurwitzian expansions 89

where α = [0; 1, a2(α), · · · ] is the RCF expansion of α.In case j ≥ i, there exists an `0 ≥ 0 such that, by Lemma 5.1 for all ` ≥ `0

T k+`p−1(x) > α ⇐⇒ i is odd,

implying ak+`p = 1 must be singularized for all ` ≥ `0 if and only if i is odd.Otherwise, they are never singularized for all ` ≥ `0.

If 1 ≤ j i, then ak+j is a constant different from aj+1(α), so

T k+`p−1(x) ≥ α ⇐⇒ j is odd and ak+j > aj+1(α).

and we see that ak+`p = 1 must be singularized for all ` > 0 if and only ifj is odd and ak+`p > aj+1(α) (or equivalently, if and only if j is even andak+`p < aj+1(α)).

Case 2: 1/2 ≤ α ≤ g. In this case we have to consider (Tk−1, Vk−1). It is clearthat there exists a minimal h ∈ 1, . . . , p such that ak+`p−h > 1 for all ` ≥ 0.If h is odd implying Vk−1 < g, then ak = 1 must be singularized if and only ifTk−1 > α. In this case, let i and j be defined as in Case 1. If j ≥ i, then thereexists an `2 ≥ 0 such that for all ` ≥ `2 one has

Tk+`p−1 > α ⇐⇒ i is odd.

If 1 ≤ j i, one has

Tk+`p−1 ≥ α ⇐⇒ j is odd and ak+j > aj+1(α).

On the other hand, if h is even implying Vk−1 > g, then ak = 1 must besingularized if and only if Tk−1 > (1− α)/α. Again let i be defined as in Case1, but j be such that

ak+j 6= aj+1((1− α)/α).

where [0; 1, a2( 1−αα ), a3( 1−α

α ), · · · ] denotes the RCF expansion of (1− α)/α.If j ≥ i, then there exists an `3 ≥ 0 such that for all ` ≥ `3 one has

Tk+`p−1 > (1− α)/α ⇐⇒ j is odd.

If 1 ≤ j i, one has

Tk+`p−1 ≥ (1− α)/α ⇐⇒ j is odd and ak+j > aj+1((1− α)/α).

2

Example 5.3 Here we give α-expansions of e for some values of α.

(i) For α = 0.7,

e = [3; −1/3, 1/2, −1/(2k + 5), 1/2]∞k=0.

90 A NOTE ON HURWITZIAN NUMBERS

(ii) For α = 0.52,

e = [3;−1/4, −1/2, 1/5,

−1/2, 1/7, −1/2, 1/9, −1/2, 1/10, 1/2, −1/(2k + 13), 1/2]∞k=0.

(iii) For α = 0.53,

e = [3;−1/4, −1/2, 1/5, −1/2, 1/6, 1/2, −1/(2k + 9), 1/2]∞k=0.

Remark 5.2 1. From the proof of Theorem 5.3 it is at once clear that x is aquadratic irrational if and only if the α-expansion of x is eventually periodic.2. Analogous to Definitions 5.1 and 5.3 we can define S-Hurwitzian numberfor any S-expansion. In case the singularization-area is ‘nice’ (such as thesingularization-areas for Nakada’s α-expansion, or for Minkowski’s diagonal con-tinued fraction expansion, see [H]), one can show that being S-Hurwitzian isequivalent to being Hurwitzian. However, it is possible to find singularization-areas S and numbers x such that x is Hurwitzian, but not S-Hurwitzian. Con-sider for example the following singularization-area S:

S =⋃

p prime

(2p + 2, 2p + 1]× (12 , 1

).

One easily convinces oneself that e does not have an S-expansion which is S-Hurwitzian.

Bibliography

[DK] Dajani, K. and Kraaikamp, C. — The Mother of All Continued Frac-tions, Coll. Math. 84/85 (2000), 109-123.

[HW] Hardy, G. H. and Wright, E. M. – An introduction to the theory ofnumbers. Fifth edition. The Clarendon Press, Oxford University Press,New York, 1979. MR 81i:10002

[H] Hartono, Y. – Minkowski’s Diagonal Continued Fraction Expansions ofHurwitzian Numbers, J. Mat. atau Pembelajarannya 8 (2002), 837–841.

[K] Kraaikamp, C. — A new class of continued fraction expansions, ActaArith. 57 (1991), no. 1, 1–39. MR 92a:11090

[KL] Kraaikamp, C. and Lopes, A. – The Theta Group and the ContinuedFraction Expansion with Even Partial Quotients, Geom. Dedicata 59(1996), no. 3, 293–333. MR 97g:58135

[N] Nakada, H. — Metrical theory for a class of continued fraction trans-formations and their natural extensions, Tokyo J. Math. 4 (1981), no.2, 399–426. MR 83k:10095

[O] Olds, C. D. – Continued fractions, Random House, New York 1963. MR26#3672

[P] Perron, O. – Die Lehre von den Kettenbruchen. Bd I. ElementareKettenbruche. (German) 3te Aufl. B. G. Teubner Verlagsgesellschaft,Stuttgart, 1954. MR 16,239e

91

92

Chapter 6

Mikowski’s DCFExpansions of HurwitzianNumbers

6.1 Introduction

A real number x with a regular continued fraction expansion of the form

[a0; a1, · · · , an, an+1(k), · · · , an+p(k)]∞k=0, (6.1)

is called Hurwitzian if a0 is an integer, ai’s are all positive integers, an+1, . . . , an+p

(called a quasi period of x) are polynomials with rational coefficients which takepositive integral values for k = 0, 1, 2, . . . and at least one of them is not con-stant. A well-known example of such numbers is e = [2; 1, 2k + 2, 1]∞k=0; see [S]for more examples.

Apart from this regular continued fraction (RCF) expansion of x there arevery many other – classical – continued fraction expansions of x; for instance,Minkowski’s diagonal continued fraction (DCF) expansion. In this paper wewill define what Hurwitzian numbers are for such continued fraction expansionsand show that their set of Hurwitzian numbers coincides with the classical setof Hurwitzian numbers.

6.2 Minkowski’s DCF

Let x ∈ [0, 1] \Q have RCF expansion

1

a1 +1

a2 +.. .

=: [0; a1, a2, · · · ]

93

94 MINKOWSKI’S DCF EXPANSIONS OF HURWITZIAN NUMBERS

and the sequence of approximation coefficients θn = θn(x), n ≥ −1, be given by

θn := Qn|Qnx− Pn|, n ≥ −1, (6.2)

where (Pn/Qn)n≥−1 is the sequence of regular convergents.In general, every x ∈ R \Q can be expanded as a continued fraction

b0 +e1

b1 +e2

b2 +.. . +

en

bn +.. .

=: [ b0; e1/b1, e2/b2, · · · , en/bn, · · · ], (6.3)

where b0 ∈ Z, x − b0 ∈ [0, 1], en = ±1, bn ∈ N and en+1 + bn ≥ 2 for n ≥1. We denote the n-th convergent of (6.3) by An/Bn. We call (6.3) a semi-regular continued fraction (SRCF) expansion of x. If, in addition, there existsan arithmetic function k : N→ N such that

An

Bn=

Pk(n)

Qk(n), n ≥ 1,

with θk(n) < 1/2, we call (6.3) the Minkowski’s diagonal continued fraction(DCF) expansion of x.

Using the expression

x =Pn + Pn−1T

nx

Qn + Qn−1Tnx

in (6.2) and putting Tn := Tnx with Tx = 1/x− b1/xc and Vn := Qn−1/Qn =[0; an, . . . , a1], one can easily see that

θn =Tn

1 + TnVn=

1an+1 + Tn+1 + Vn

, n ≥ 0. (6.4)

Since θn > 1/2 implies Tn > 1/2 and hence an+1 = 1, it was shown in [K]that DCF expansions can be obtained from RCF expansions by singularizingan+1 = 1. We see that the DCF expansion can be viewed as an S-expansionwith singularization area

SDCF =

(T⋃

, V ) ∈ [0, 1)× [0, 1] :T

1 + TV> 1/2

;

see [K] for details. This singularization area SDCF is the area under the curvev = 2−1/t in Figure 6.1. The singularization process is based upon the identity

A +e

1 +1

B + ξ

= A + e +− e

B + 1 + ξ.

An equivalent algorithm to transform the RCF expansion of x into the cor-responding DCF expansion of x is given in [K]. In this algorithm only the first

DCF-Hurwitzian expansion 95

Figure 6.1: DCF Singularization Area

and the last 1’s in a block of consecutive 1’s may or may not be singularized.As a consequence of this algorithm, Minkowski’s DCF expansion of a quadraticirrational is periodic. In the next section we will define Hurwitzian numbers forMinkowski’s DCF expansions and show that any Hurwitzian number is DCF-Hurwitzian. Conversely, the DCF expansion of a Hurwitzian number is againHurwitzian.

6.3 DCF-Hurwitzian expansion

Definition 6.1 Let x ∈ R \Q. Then x has a DCF-Hurwitzian expansion if

x = [ b0; e1/b1, · · · , en/bn, en+1/bn+1(k), · · · , en+p/bn+p(k) ]∞k=0 (6.5)

where b0 ∈ Z, x − b0 ∈ [0, 1], en = ±1, bn ∈ N and en+1 + bn ≥ 2 for n ≥ 1.Moreover, for i = 1, . . . , p we must have that bn+i are polynomials with rationalcoefficients which take positive integral values for k = 0, 1, 2, . . ., and at leastone of them is non-constant.

Notice that the sequence of convergents of (6.5) forms a subsequence ofregular convergents with approximation coefficients less than 1/2.

The following lemma is very useful.

Lemma 6.1 Let x, y ∈ [0, 1] with RCF expansions

x = [0; a1(x), a2(x), · · · ], y = [0; a1(y), a2(y), · · · ].Let x 6= y and k ∈ N be such that

a1(x) = a1(y), . . . , ak−1(x) = ak−1(y), and ak(x) 6= ak(y).

96 MINKOWSKI’S DCF EXPANSIONS OF HURWITZIAN NUMBERS

Then one has

x > y if and only if

ak(x) < ak(y) if k is odd

ak(x) > ak(y) if k is even

We now prove the main result in this section.

Theorem 6.1 Let x ∈ R \ Q. Then x is Hurwitzian if and only if x has aDCF-Hurwitzian expansion.

Proof. Let x be a Hurwitzian number (6.1). Let m0 ∈ N be such that m0 > nand for all m ≥ m0 all non-constant polynomials am are greater than 1 andmonotonically increasing (i.e., am < am+p < am+2p < · · · ). Furthermore, let

k0 = mink ≥ m0 : ak > 1

and i ∈ 1, . . . , p−1 be such that ak0+i = 1 (If such an i does not exist, we aredone!). Since only the first and the last 1’s of a block of consecutive 1’s may besingularized we first assume that ak0+i = 1 is the first 1 in a block of 1’s, i.e.,ak0+i−1 ≥ 2. We denote by λ ∈ 1, . . . , p − 1 the length of the block of 1’s,that is,

ak0+i−1 6= 1, ak0+i = ak0+i+1 = · · · = ak0+i+λ−1 = 1, ak0+i+λ 6= 1.

We now consider the following cases.

(A) λ = 1 (area A in Figure 6.1): ak0+i will always be singularized.

(B) ak0+i−1 = 2 and λ = 2 (area B in Figure 6.1): ak0+i will never be singular-ized.

(C) ak0+i−1 ≥ 3 and λ ≥ 3 (area C in Figure 6.1): ak0+i will always be singu-larized.

(D) ak0+i−1 ≥ 3 and λ = 2 (area D in Figure 6.1): ak0+i may or may not besingularized.

(E) ak0+i−1 = 2 and λ ≥ 3 (area E in Figure 6.1): ak0+i may or may not besingularized.

In general, let n` = k0 + i + `p − 1. Then, for all ` ∈ N, due to the quasiperiodicity an`+1 will always be singularized in cases A and C and will neverbe singularized in case B. We now look at cases D and E more carefully. Noticefirst that (6.4) yields

(Tn`, Vn`

) ∈ SDCF ⇐⇒ Tn`+1 < 1− Vn`,

DCF-Hurwitzian expansion 97

or, equivalently,

[0; an`+2, · · · ] < [0; 1, an`− 1, an`−1, · · · , a1]. (6.6)

In case D we have for each `

Tn`+1 = [0; 1, an`+3, · · · ],Vn`

= [0; an`, an`−1, · · · , a1].

Now, by Lemma 6.1, since an`+2 = 1, (6.6) holds if

an`+3 < an`− 1.

This leads to the following possibilities.

(i) an`+3 and an`are both constant polynomials: an`+1 = 1 will always be

singularized for all ` ∈ N in case an`+3 < an`−1; and never be singularized

in case an`+3 > an`− 1.

(ii) an`+3 is constant, but an`non-constant: there exists an `1 ∈ N such that

an`+1 = 1 will always be singularized for all ` ≥ `1.

(iii) an`+3 is non-constant, but an`constant: there exists an `2 ∈ N such that

an`+1 = 1 will never be singularized for all ` ≥ `2.

(iv) an`+3 and an`are both non-constant polynomials: there exists an `3 ∈ N

such that an`+1 = 1 will either always or never be singularized for all` ≥ `3.

In case an`+3 = an`− 1, (6.6) holds if an`+4 > an`−1, and we can again discuss

four possible cases, etc. (Note that by definition of m0 we at least have tocompare at most 2p pairs of digits.)

In case E we have for each `

Tn`+1 = [0; 1λ−1, an`+3, · · · ],Vn`

= [0; 2, an`−1, · · · , a1].

where 1c is an abbreviation of 1, · · · , 1︸ ︷︷ ︸c−times

.

Now it is easy to see that (6.6) holds if

[0; 1λ−3, an`+λ+1, · · · ] < [0; an`−1, · · · , a1],

and a reasoning similar to that in case D shows that there exists an `0 such thatan`+1 = 1 will either always or never be singularized for all ` ≥ `0.

Thus we see that the first 1 in a block of λ consecutive ones will either alwaysor never be singularized from a certain index on. This is also true for the last 1in the block, that is an`+λ = 1, in all cases with λ 6= 1 (In case λ = 1 the last

98 MINKOWSKI’S DCF EXPANSIONS OF HURWITZIAN NUMBERS

1 is also the first 1, and will always be singularized). First notice that for thiscase we have

Tn`+λ = [0; an`+λ+1, · · · ],Vn`+λ−1 = [0; 1λ−1, an`

, · · · , a1],

and that (6.4) yields

(Tn`+λ−1, Vn`+λ−1) ∈ S ⇐⇒ Vn`+λ−1 < 1− Tn`+λ,

which is by Lemma 6.1 equivalent to

[0; 1λ−2, an`, · · · , a1] > [0; an`+λ+1 − 1, · · · ].

The proof now follows from arguments similar to those used the above casesfor the first 1 in a block of λ consecutive ones. For instance, if an`+λ+1 > 2,an`+λ = 1 will always be singularized.

Finally, if x has a DCF-Hurwitzian expansion, then the (quasi) periodicityof this expansion together with the fact that the singularization process can bereversed only in a unique way yields that x is Hurwitzian, which completes theproof. 2

Acknowledgement

The author would like to express his gratitude to Cor Kraaikamp for his valuablecomments and ideas.

Bibliography

[K] Kraaikamp, C. – Statistic and Ergodic Properties of Minkowski’s Diago-nal Continued Fraction, Theoret. Comp. Sci. 65 (1989) no. 2, 197–212.MR 90m:11120

[S] Stambul, P. – A generalization of Perron’s theorem about Hurwitziannumbers, Acta Arith. 80 (1997), 141–148. MR 98h:11013

99

100

Samenvatting

Dit proefschrift bestaat uit 5 artikelen, die verschillende aspecten van de theorieder kettingbreuken behandelen.

In hoofdstuk 21 worden verschillende kettingbreuk algorithmen met oneven wi-jzergetallen bestudeerd, in het bijzonder de klassieke Odd Continued Fraction(OddCF) expansion, and Rieger’s Grotesque Continued Fraction (GCF). Derelatie tussen de OddCF en de reguliere kettingbreuk (RCF) wordt gegevenaan de hand van de singularizatie en insertie processen. Gebruikmakend vanSchweiger’s natuurlijke uitbreiding van het ergodische systeem dat ‘onder’ deOddCF ligt wordt aangetoond dat de rij van convergenten van de kettingbreuknaar dichtsbijzijnde gehele (NICF) altijd een deelrij is van de overeenkomstige rijvan de convergenten van de OddCF. Gebruikmakend van methodes ontwikkeldin in [JK] wordt het Tong pre-spectrum voor de OddCF bepaald, zie ook [T1]voor Tong’s oorspronkelijke resultaat voor de NICF.

Verder wordt aangetoond dat de Grotesque Continued Fraction ook metbehulp van singularizaties en inserties verkregen kan worden uit de RCF. Inzekere zin hebben de GCF en Hurwitz’ singular continued fraction expansiondezelfde onderlinge relatie als de OddCF en de NICF hebben.

Tenslotte worden nog verschillende maximale en niet-maximale kettingbreukalgorithmen met oneven wijzergetallen beschouwd.

In hoofdstuk 32 wordt de afbeelding TE : [0, 1) → [0, 1) bestudeerd, gedefinieerddoor

TE(x) :=1b 1

xc

(1x− b 1

xc)

, x 6= 0; TE(0) := 0.

Door deze afbeelding te itereren vinden we voor elke x ∈ [0, 1) een unieke

1Gezamelijk werk met Cor Kraaikamp, dat zal verschijnen in Rev. Romaine Math. PuresAppl. 47 (2002), no. 1.

2Gezamelijk werk met Cor Kraaikamp en Fritz Schweiger, dat zal verschijnen in J. deTheorie des Nombres de Bordeaux 14 (2002).

101

102 Samenvatting

kettingbreuk-ontwikkeling met niet-dalende wijzergetallen, van de vorm

1

b1 +b1

b2 +b2

b3 +.. . +

bn−1

bn +.. .

, bn ∈ N, with bn ≤ bn+1 .

Deze kettingbreuk ontwikkeling noemen we—in analogie met de klassieke Engelreeks ontwikkeling—de Engel kettingbreuk (ECF) ontwikkeling van x.

De ECF heeft een aantal basis eigenschappen gemeen met de RCF, maaris op een aantal punten opvallend verschillend van de RCF (of willekeurig elkeander semi-reguliere kettingbreuk). In het bijzonder gedragen de convergentenvan de ECF zich verschillend.

Er wordt bewezen dat TE ergodies is m.b.t. de Lebesgue maat λ, en dat ergeen eindige invariant maat bestaat, equivalent met de Lebesgue maat. Verderwordt er aangetoond dat er voor de afbeelding TE oneindig veel σ-eindige,oneindige invariante maten bestaan. Twee verschillende families van dergeli-jke maten worden gegeven.

Tenslotte wordt er aangetoond dat de ECF isomorf is met Ryde’s monotonen,nicht-abnehmenden Kettenbruch (MNK). Deze MNK heeft als onderliggendetransformatie de afbeelding TR : ( 1

2 , 1) → (12 , 1), gedefinieerd door:

TR(x) = SR(x) =k

x− k, for x ∈ R(k) :=

(k

k + 1,k + 1k + 2

), k ∈ N.

Dit isomorfisme heeft als direct gevolg dat alle metrische resultaten voor deECF ook voor de MNK bestaan, en omgekeerd, dat alle bekende arithmetischeresultaten voor de MNK kunnen worden overgevoerd naar de ECF. Zo bestaaner kwadraties irrationale getallen waarvoor de MNK-ontwikkeling niet eventueelperiodiek zijn, en voor dezelfde kwadraties irrationale getallen zijn de wijzerge-tallen van de ECF-ontwikkeling dus onbegrensd.

Het Hurwitz-spectrum voor de kettingbreuk naar dichtsbijzijnde gehele (NICF)werd voor het eerst in [JK] bestudeerd door Jager en Kraaikamp. Als we deNICF approximatie coefficienten noteren door (Θn)n≥1, dan geldt er dat

min(Θn−1, Θn, Θn+1) <52(5√

5− 11) = 0.4508 · · · .

Bovendien toonden Jager en Kraaikamp aan dat voor elk irrationaal x er geldtdat Θn < 1/

√5 voor oneindig veel n ≥ 1. In [T1] en [T2] werden deze resultaten

door Tong verdiept. Tong bewees dat

min(Θn−1, Θn, . . . , Θn+k) <1√5

+1√5

(3−√5

2

)2k+3

, k ≥ 1.

Samenvatting 103

Hoofstuk 43 geeft een nieuw bewijs van Tong’s resultaat, uitgaande van demethode van [JK]. Hierdoor is het bewijs niet alleen eenvoudiger en toepasbaarop andere kettingbreuk algorithmes (zoals Nakada’s α-expansions), maar kun-nen er ook verschillen metrische conclusies getrokken worden voor bijna alle xm.b.t. Tong’s spectrum.

Een getal x ∈ R noemen we Hurwitzian als de reguliere kettingbreuk ontwikke-ling van x (1.2) geschreven kan worden als

x = [a0; a1, · · · , an, an+1(k), · · · , an+p(k)]∞k=0,

waarbij an+1(k), . . . , an+p(k) (de zogenaamde quasi periode van x) polynomenzijn met rationale coefficienten die postive gehele waarden aannemen voor k =0, 1, 2, . . ., en er bovendien geldt dat minstens een van deze polynomen niet con-stant is. Het concept van Hurwitzian getallen is een generalizatie van getallenmet een uiteindelijk periodieke (reguliere) kettingbreuk ontwikkeling. In hoofd-stuk 54 definieren we Hurwitzian getallen voor de NICF, de ‘backward’ ketting-breuk ontwikkeling, en voor Nakada’s α-expansions. Er wordt aangetoond datde verzamelingen van Hurwitzian getallen voor de genoemde kettingbreuk algo-rithmen samenvallen met de klassieke verzameling van Hurwitzian getallen. Debewijzen rusten op de vorm van de singularizatie-gebieden van de genoemde ket-tingbreuk algorithmen. Er wordt verder aangetoond dat er S-expansies bestaanwaarbij de verzameling van Hurwitzian getallen niet met de klassieke verzamel-ing samenvalt.

Hoofdstuk 65 is een voortzetting van het voorafgaande hoofdstuk, hoofdstuk 5.In dit hoofdstuk worden Hurwitzian getallen voor Minskowski’s diagonal contin-ued fraction (DCF) expansion gedefinieerd. Ook voor deze kettingbreuk valt decollectie Hurwitzian getallen samen met de klassieke verzameling van Hurwitziangetallen. De situatie is voor de DCF ingewikkelder dan voor de kettingbreukendie in het vorige hoofdstuk beschouwd werden, omdat het singularizatie gebieddat de DCF karakteriseerd NICF ingewikkelder is.

3Gezamelijk werk met Cor Kraaikamp4Gezamelijk werk met Cor Kraaikamp in Tokyo J. Math. 25 (2002), no. 25J. Matematika atau Pembelajarannya VIII (2002), 837–841.

104 Samenvatting

Bibliography

[B] Bressoud, D. M. – Factorization and Primality Testing, Sringer-Verlag,New York, (1989). MR 91e:11150

[BJW] Bosma, W., H. Jager and F. Wiedijk. - Some metrical observations onthe expansion by continued fractions, Indag. Math. 45 (1983), 353–379.

[DK] Dajani, K. and C. Kraaikamp. – Ergodic Theory of Numbers, Carus Math-ematical Monographs, No. 29, (2002).

[Di] Dickson, L. E. – History of the Theory of Numbers, Vols I, II, III, CarnegieInstitution of Washington, Washington, (1991-1923).

[Do] Doeblin, W. — Remarque sur la theorie metrique des fractions continues,Compositio Math. 7 (1940), 353–371. MR 2,107e

[HW] Hardy, G. H. and Wright, E. M. – An introduction to the theory ofnumbers. Fifth edition. The Clarendon Press, Oxford University Press,New York, (1979). MR 81i:10002

[IK] Iosifescu, M. and C. Kraaikamp. - The Metrical Theory of ContinuedFractions, Kluwer Academic Press, Dordrecht, The Netherlands, (2002).

[J] Jager, H. - Continued fraction and ergodic theory, Transcendental Num-bers and Related Topics, RIMS Kokyuroku, 599, Kyoto University, Ky-oto, Japan, (1986), 55–59.

[JK] Jager, H. and C. Kraaikamp. — On the approximation by continued frac-tions, Indag. Math. 51 (1989), no. 2, 289-307. MR 90k:11084

[Kh] Khintchine, A. Ya. - Metrische kettenbruchprablemen , Compositio Math.1 (1935), 361–382.

[Kn] Knuth, D. E. - The distribution of continued fraction approximations, J.Number Theory 19 (1984), no 3, 443–448. MR 86d:11058

[K1] Kraaikamp, C. - Metric and Arithmetic Results for Continued FractionExpansions, Ph.D. Thesis (1990), Universiteit van Amsterdam, Amster-dam.

105

106 BIBLIOGRAPHY

[K2] Kraaikamp, C. - A new class of continued fraction expansions, Acta Arith.57 (1991), no. 1, 1–39. MR 92a:11090

[Ku] Kuzmin, R. O. - On a problem of Gauss , Dokl. Akad. Nauk. SSSR Ser.A, (1928), 375–380.

[N] Nakada, H. — Metrical theory for a class of continued fraction transfor-mations and their natural extensions, Tokyo J. Math. 4 (1981), no. 2,399–426. MR 83k:10095

[NIT] Nakada, H., S. Ito and S. Tanaka — On the invariant measure for thetransformations associated with some real continued fractions, Keio EngrgRep., 30 (1977), no. 13, 159–175. MR 58 16574

[O] Olds, C. D. – Continued fractions, Random House, New York, (1963).MR 26#3672

[P] Petersen, K. – Ergodic Theory, Cambridge University Press, Cambridge,(1997).

[R-N] Ryll-Nardzewski, C. - On the ergodic theorems. II. Ergodic theory of con-tinued fractions, Studia Math. 12 (1951), 74–79.

[S] Schweiger, F. – Ergodic theory of fibred systems and metric number theory,Oxford Science Publications. The Clarendon Press, Oxford UniversityPress, New York, (1995). MR 97h:11083

[T1] Tong, Jing Cheng – The conjugate property of the Borel theorem onDiophantine approximation, Math. Z. 184 (1983), no. 2, 151–153. MR85m:11039

[T2] Tong, Jing Cheng — Approximation by nearest integer continued fractions(II), Math. Scand. 74 (1994), no. 1, 17–18. MR 95c:11085

[V] Vahlen, K. Th. — Uber Naherungswerte und Kettenbruche, Journal f. d.reine und angew. Math. 115 (1895), 221–233.

[Wa] Walters, P. – An Introduction to Ergodic Theory, Springer-Verlag NewYork, Inc., New York, (2000).

[Wi] Wirsing, E. – On the theorem of Gauss-Kuzmin-Lery and a Frobeniustype theorem for function spaces, Acta Arith. 24 (1974), 507–528. MR 492637

Curriculum Vitae

there are three things that last for ever:faith, hope, and love;

but the greatest of them all is love.

1 Cor 13:13

Yusuf Hartono was born on November 16, 1964 inKundur, a leprosy colony near Palembang, South Sumatra,Indonesia. He started his elementary education at SekolahDasar Negeri 3, Mariana in 1971, but graduated from Seko-lah Dasar Negeri 2, Mariana in 1976. He got his next three-year secondary education from SMP Bina Utama, SungaiGerong. Afterwards, he spent another three years at highschool SMA Yaktapena 2, Sungai Gerong. After graduating

from high school in 1983, he went to Universitas Sriwijaya in Palembang, fromwhich he obtained his bachelor’s degree Sarjana in Mathematics Education fiveyears later. In 1990 he started his job as a junior lecturer at the Department ofMathematics Education Universitas Sriwiyaja. He then spent two years study-ing at the Department of Mathematics and Computer Science University ofMissouri-Rolla, USA, and obtained Master of Science in Applied Mathemat-ics in 1993. In October 1998 he joined Afdeling CROSS (Control, Risk, Opti-mization, Systems and Stochastics) Faculteit Informatietechnologie en Systemen(ITS) Technische Universiteit Delft, the Netherlands and started his Ph.D. re-search in 1999 under supervision of Cor Kraaikamp. In the period of 1993–1998he gave lectures in Statistics and Educational Research Methodology.

107