Hurwitz’s Complex Continued Fractions A Historical Approach and Modern Perspectives. Dissertationsschrift zur Erlangung des naturwissenschaftlichen Doktorgrades der Julius-Maximilians-Universit¨at W¨urzburg vorgelegt von Nicola Oswald aus Kronach W¨urzburg 2014 2 ”Knowledge is in the end based on acknowledgement.” Ludwig Wittgenstein (1889 - 1951) Ich freue mich, an dieser Stelle die Gelegenheit zu nutzen, einigen Menschen zu danken, die mich w¨ahrend der letzten drei Jahre unterst¨utzt und begleitet haben. Mein erstes großes Dankesch¨on geb¨uhrt nat¨urlich J¨orn Steuding, der mir diese Promotionsstelle nicht nur erm¨oglicht hat, sondern mir seitdem als Betreuer stets mit Rat und Tat zur Hilfe steht. Er erm¨oglichte mir den Besuch zahlreicher Konferenzen, meine Forschungsinter- essen zu finden und selbstst¨andig eigene Projekte zu verwirklichen. In unserer gemein- samen Arbeitsgruppe zusammen mit Thomas Christ, sowie am Institut, genoss ich sehr die fortw¨ahrend freundliche und r¨ucksichtsvolle Atmosph¨are. Außerdem m¨ochte ich meinen Dank Karma Dajani aussprechen, die mir bereits zu Beginn meiner Arbeit erste Inspirationen zur Ergodentheorie gab. W¨ahrend ihres Aufen- thalts in W¨urzburg im Sommersemester 2014 hatte ich große Freude daran, dieses Inter- esse wieder aufzunehmen und konnte viel pers¨onliche Motivation aus ihrer herzlichen Art sch¨opfen. Im Bezug auf meine mathematik-historische Forschung m¨ochte ich mich ausdr¨ucklich bei meiner Mentorin Renate Tobies und Klaus Volkert bedanken. Beide halfen mir sehr bei meinem Einstieg in diesen neuen Fachbereich und ich bin ihnen f¨ur viele gute Ratschl¨age und Ermutigungen dankbar. In dieser Hinsicht sind auch das Archiv der ETH Z¨urich, insbesondere die hervorra- gende Betreuung von Evelyn Boesch, die Staats- und Universit¨atsbibliothek G¨ottingen, das Staatsarchiv Basel und das Universit¨atsarchiv Halle-Wittenberg zu erw¨ahnen. Ich bedanke mich herzlich f¨ur die M¨oglichkeit, so viele spannende Dokumente einsehen zu d¨urfen. Zu guter Letzt m¨ochte ich mich von ganzem Herzen bei Ingrid und Wolfgang, Moni und Nelly, Eva, Daria, Aniela, Lisa und nat¨urlich Andi bedanken, die stets daf¨ur gesorgt haben, dass meine Nerven wieder an ihren angedachten Ort wandern. 3 4 Contents Introduction....................................... 8 1 Real Continued Fractions 11 1.1 A Brief Mathematical and Historical Introduction . .......... 11 1.2 SomeClassicalResults. 16 2 The Hurwitz Brothers 19 2.1 TheYounger:AdolfHurwitz . 20 2.2 Excursion: The Personal Hurwitz Estate from the Archive of the ETH Zurich 31 2.2.1 Recreational Mathematics in the Mathematical Diaries........ 34 2.2.2 Adolf Hurwitz Folding and Cutting Paper . 44 2.2.3 Relation to his Student David Hilbert . 53 2.3 TheElderBrother: JuliusHurwitz . 76 2.4 Excursion: A Letter Exchange Concerning Julius Hurwitz’s PhD Thesis . 85 3 Hurwitz’s Approach to Complex Continued Fractions 95 3.1 Continued Fractions According to Adolf . ...... 95 3.2 A Complex Continued Fraction According to Julius . ........106 3.3 SomeHistoricalNotes ............................. 113 4 Modern Developments of Complex Continued Fractions 117 4.1 J. Hurwitz’s Algorithm = Tanaka’s Algorithm . .......118 5 Contents 4.1.1 Basics ...................................119 4.1.2 Example..................................122 4.1.3 Some Considerations and Characteristics . 124 4.1.4 Geometrical Approach to the Approximation Behaviour .......132 4.1.5 ApproximationQuality . .135 4.2 ErgodicTheory..................................139 4.2.1 Transformations .............................139 4.2.2 Continued Fraction Transformation . 140 4.2.3 Dual Transformation and Natural Extension for Tanaka’s Algorithm 142 4.2.4 Variation on the D¨oblin-Lenstra Conjecture . .......146 4.3 TranscendentalNumbers. 151 4.3.1 Roth’sTheorem .............................152 4.3.2 Transfer to Tanaka’s Continued Fraction . 153 4.3.3 Example of a Transcendental Number . 155 5 Representation of Real Numbers as Sum of Continued Fractions 159 5.1 IntroductionandMainResult. 159 5.2 SomePreliminaries............................... 161 5.3 ACusickandLee-typetheorem. 165 5.4 TransfertotheComplexCase. 170 6 A New Type of Continued Fractions with Partial Quotients from a Lattice 173 6.1 Cyclotomic Fields: Union of Lattices . 173 6.2 Generalization: Vorono¨ıDiagrams . 181 7 Approximation Quality 185 7.1 PreviousResults ................................. 186 7.2 MinkowskiandGintner ............................. 187 7.2.1 Preliminaries ...............................188 6 Contents 7.2.2 Application of Linear Forms . 189 7.2.3 Improvement by the Lattice Point Theorem . 191 7.2.4 Geometrical Interpretation . 193 7.2.5 Generalization of the Result . 194 7.3 ModularGroup..................................198 7.4 Arithmetical Application . 201 7.5 LawofBestApproximation . 205 8 Last words 207 9 Appendix 209 9.1 Appendix I: Recreational Mathematics in the Mathematical Diaries . 209 9.2 Appendix II: Links to Hilbert in the ETH Estate of Hurwitz .........210 9.3 AppendixIII:TableofFigures . 212 7 Introduction This thesis deals with two branches of mathematics: Number Theory and History of Mathematics. On the first glimpse this might be unexpected, however, on the second view this is a very fruitful combination. Doing research in mathematics, it turns out to be very helpful to be aware of the beginnings and development of the corresponding subject. In the case of Complex Continued Fractions the origins can easily be traced back to the end of the 19th century (see [Perron, 1954, vl. 1, Ch. 46]). One of their godfathers had been the famous mathematician Adolf Hurwitz. During the study of his transformation from real to complex continued fraction theory [Hurwitz, 1888], our attention was arrested by the article ’Ueber eine besondere Art der Kettenbruch-Entwicklung complexer Gr¨ossen’ [Hurwitz, 1895] from 1895 of an author called J. Hurwitz. We were not only surprised when we found out that he was the elder unknown brother Julius, furthermore, Julius Hurwitz introduced a complex continued fraction that also appeared (unmentioned) in an ergodic theoretical work from 1985 [Tanaka, 1985]. Those observations formed the basis of our main research questions: What is the historical background of Adolf and Julius Hurwitz and their mathematical studies? and What modern perspectives are provided by their complex continued fraction expansions? In this work we examine complex continued fractions from various viewpoints. After a brief introduction on real continued fractions, we firstly devote ourselves to the lives of the brothers Adolf and Julius Hurwitz. Two excursions on selected historical aspects in respect to their work complete this historical chapter. In the sequel we shed light on Hur- witz’s, Adolf’s as well as Julius’, approaches to complex continued fraction expansions. Correspondingly, in the following chapter we take a more modern perspective. Highlights are an ergodic theoretical result, namely a variation on the D¨oblin-Lenstra Conjecture [Bosma et al., 1983], as well as a result on transcendental numbers in tradition of Roth’s theorem [Roth, 1955]. In two subsequent chapters we are concernced with arithmetical properties of complex continued fractions. Firstly, an analogue to Marshall Hall’s theo- rem from 1947 [Hall, 1947] on sums of continued fractions is derived. Secondly, a general approach on new types of continued fractions is presented building on the structural prop- erties of lattices. Finally, in the last chapter we take up this approach and obtain an upper bound for the approximation quality of diophantine approximations by quotients of lattice points in the complex plane generalizing a method of Hermann Minkowski, improved by Hilde Gintner [Gintner, 1936], based on ideas from geometry of numbers. In particular in Chapter 2 the reader can find a great number of quotations. Notice that all translations from German to English have been made by the author to the best of her knowledge. The original versions can be found in footnotes. Whenever a result of us or a related version has already been published, the correspond- ing reference is given in each chapter’s preamble. At this point, we align ourself to an opinion of the authors of the inspiring book ’Nev- erending Fractions’ [Borwein et al., 2014, p. i]: ”[D]espite their classical nature, continued fractions are a neverending re- search area [...]”. 1 Real Continued Fractions This chapter shall provide a brief mathematical as well as historical introduction to the theory of real continued fractions. Since there is a great variety of excellent standard literature, here we avoid to repeat comprehensive proofs. Instead, our aim is to anticipate some results which will occur in subsequent chapters for the complex relatives of real continued fractions. To achieve a deeper understanding and an extensive overview of real continued fraction theory, as well as for references to the original papers, among others we refer to [Perron, 1913], [Opolka and Scharlau, 1980], [Kraaikamp and Iosifescu, 2002], [Steuding, 2005], and [Borwein et al., 2014]. 1.1 A Brief Mathematical and Historical Introduction Every real number x can be uniquely expanded as 1 x = a + , 0 1 a1 + a + . 2 .. 1 + an + . .. where a Z and ai N; if x is rational, the expansion is finite and uniqueness follows 0 ∈ ∈ from assuming the last an to be strictly larger than one. This often with a twinkling eye called typographical nightmare represents the regular continued fraction expansion of x. It is developed