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Continued Fractions and Hyperbolic Geometry Open Research Online The Open University’s repository of research publications and other research outputs Continued Fractions and Hyperbolic Geometry Thesis How to cite: Walker, Mairi (2016). Continued Fractions and Hyperbolic Geometry. PhD thesis The Open University. For guidance on citations see FAQs. c 2016 The Author https://creativecommons.org/licenses/by-nc-nd/4.0/ Version: Version of Record Link(s) to article on publisher’s website: http://dx.doi.org/doi:10.21954/ou.ro.0000ef81 Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online’s data policy on reuse of materials please consult the policies page. oro.open.ac.uk CONTINUED FRACTIONS AND HYPERBOLIC GEOMETRY MAIRI WALKER, MMATH Department of Mathematics and Statistics The Open University A thesis submitted for the degree of Doctor of Philosophy D e c e m b e r 24th 2015 bAT£ of so £i y m s s to A : 2 3 C ^ c e r v \6 < o A - a «j rVP(g.i ^ ( ProQuest Number: 13834780 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 13834780 Published by ProQuest LLC(2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 ABSTRACT This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. This intuitive approach gives insight into the theory of continued fractions that is not so easy to ob­ tain from traditional algebraic methods. Using it, we provide a more extensive study of both Rosen continued fractions and even-integer continued fractions than in any previous works, yielding new results, and revisiting classical theorems. We also study two types of complex continued fractions, namely Gaussian integer continued fractions and Bianchi continued fractions. As well as providing a more elegant and simple theory of continued fractions, our approach leads to a natu­ ral generalisation of continued fractions that has not been explored before. 'Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real?' — Albus Dumbledore1 This thesis is dedicated to my mother, Dr. Jill Walker, who set the ball rolling fifteen years ago when she showed me her own PhD thesis, and gave me the desire to discover something that nobody else has ever discovered before. i in J. K. Rowling's Harry Potter and the Deathly Hallows DECLARATION I confirm that the material contained in this thesis is the result of inde­ pendent work. None of it has previously been submitted for a degree or other qualification to this or any other university or institution. Material from Chapter 2 and Chapter 3 appears in the following two papers. [54] Short, I. and Walker, M., Even-integer continued fractions and the Farey tree, accepted for publication. ArXiv:i5o8.oi373. [53] Short, I. and Walker, M., Geodesic Rosen continued fractions, submitted for publication. ArXivn^io.i^S^vz. December 24th 2015 Mairi Walker ACKNOWLEDGEMENTS I begin with a huge thank you to my supervisor Ian Short, who in­ troduced me to the beauty of the topic of this thesis, supported me throughout my time at The Open University, and encouraged me to make the most of every opportunity present to me as a PhD student. I could not have wished for a better supervisor. Thanks go also to the many other members of staff in the Department of Mathematics and Statistics who have made my time there a pleasure. I am indebted too to Meira Hockman from the University of the Witwatersrand for kindly hosting my research visit there from April to June 2015. Special thanks go to my officemates Rosie, Matthew, David and Vasso, who have been so overwhelmingly generous with their advice, support and time that I don't think I'll ever be able to thank them enough. To Ian Allen too, who has always been there to celebrate with me on the good days, and to wipe away the tears and pour the gin on the bad days. ix CONTENTS 1 INTRODUCTION 1 1.1 Preliminaries 2 1.2 Continued fractions and hyperbolic geometry 7 1.3 Outline of thesis 17 2 ROSEN CONTINUED FRACTIONS 21 2.1 Rosen continued fractions and hyperbolic geometry 22 2.2 Geodesic Rosen continued fractions 31 2.3 The number of geodesic Rosen continued fractions 43 2.4 Characterising geodesic Rosen continued fractions 66 2.5 Infinite Rosen continued fractions 8i 3 EVEN-INTEGER CONTINUED FRACTIONS 85 3.1 Even-integer continued fractions and hyperbolic geom­ etry 86 3.2 The theory of even-integer continued fractions 96 3.3 Even-integer continued fractions and Diophantine ap­ proximation 118 3.4 Rosen continued fractions when A > 2 133 4 FURTHER TOPICS ON CONTINUED FRACTIONS & HYPER­ BOLIC GEOMETRY 1 3 7 4.1 Gaussian integer continued fractions 138 4.2 Bianchi continued fractions 154 4.3 Semigroups of Mobius transformations 164 BIBLIOGRAPHY 179 INTRODUCTION It is well-known that a continued fraction can be studied using se­ quences of Mobius transformations, and that by using the action of Mobius transformations on hyperbolic space, continued fractions can be represented geometrically in a number of ways. However, little attempt has been made to use these geometric representations to de­ velop a theory of continued fractions, and there are many classes of continued fractions for which no such geometric treatment has been explored. This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. Our intuitive approach gives us insight into the theory of continued fractions that is not so _ easy to obtain from traditional algebraic methods. It allows us to pro­ vide a more extensive study of Rosen continued fractions and even- integer continued fractions than in any previous works, yielding new results, and revisiting classical theorems. We also use it to study two types of complex continued fractions. As well as providing a more simple and elegant theory of continued fractions, our approach leads to a natural generalisation of continued fractions that has not been explored before. The remainder of this chapter introduces the background material upon which this thesis is built. INTRODUCTION 1.1 PR ELIM IN A R IES We begin by introducing the preliminary material on continued frac­ tions, hyperbolic geometry and graph theory that is used throughout this thesis. 1.1.1 CONTINUED FRACTIONS A continued fraction is an expression of the form ai b i + -----------: a2 b2 + b3 H----- where the coefficients at and bt may be any real or complex numbers with at 7^ 0 for i = 1, 2, This thesis concerns continued fractions of the form 1 b i + --------------------------------, (1.1) where the coefficients bt belong to some set of real or complex num­ bers. Such a continued fraction may be infinite or finite, .depending on whether the sequence bt is infinite or finite respectively. We de­ note the continued fraction by [bi, b2,... ] in the former case, and by [b i,..., bn] in the latter. A regular continued fraction is a continued fraction in which all the coefficients bt are positive integers, except bi which may be zero or a negative integer. Although regular continued fractions are the focus of the classical theory of continued fractions, many other classes of continued fractions have been studied extensively; several of these 1.1 PRELIMINARIES are discussed in this thesis, as well as some other classes of continued fractions that have received relatively little attention. The value of a finite continued fraction [bi,... ,brJ is the number obtained by evaluating that expression. Sometimes we abuse notation and use [b i,..., bn] to represent the value of [b i,..., bn]. This is quite natural; the distinction between continued fractions and their values is blurred in most works on continued fractions. The convergents of a finite or infinite continued fraction are the continued fractions ck = [bi,...,bk], for k = 1, 2, Again, we will sometimes use the term 'convergent' to refer to the value of a convergent. If the sequence of convergents of an infinite continued fraction converges to a point x in C U {oo}, then we say that the continued fraction converges and has value x. If the continued fraction [bi, b2,... ] has value x, then we also say that it is a continued fraction expansion ofx. We will often find it more convenient to work with 'minus' contin­ ued fractions of the form 1 b i ---------------------- . (1.2) 1 rather than the 'plus' continued fractions of the form shown in Equa­ tion (i.i). In a sense, the two types of continued fractions are equiva­ lent, because we can move from one to the other by changing the sign of the coefficients bi when i is even. We use the same notation and terminology for minus continued fractions (such as [bi,b2,...] and so forth) as we do for plus continued fractions. INTRODUCTION y 1.1.2 HYPERBOLIC GEOMETRY Throughout this thesis we will, unless otherwise specified, use the upper half-plane model of two-dimensional hyperbolic space, H , which is the set H = {z = x + y ie C | y > 0}, equipped with the infinitesimal metric , 2 dx2 + dy2 ds ------- = -^ -.
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