Nicola Oswald

• The thesis ’Hurwitz’s Complex Continued Fractions - A Historical Approach and Modern Perspectives.’ 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 (seeThe thesis ’Hurwitz’s Complex Continued Fractions - A Historical Approach and Modern Perspectives.’ 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 Grössen’ [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 Hurwitz’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 Döblin-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 Theorem 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 properties 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.…
• Die Arbeit ’Hurwitz’s Complex Continued Fractions - A Historical Approach and Modern Perspectives.’ beschäftigt sich übergreifend mit zwei Teilbereichen der Mathematik: Zahlentheorie und Geschichte der Mathematik. Im ersten Teil wird ein historischer Blick auf das Leben und Wirken der Gebrüder Adolf und Julius Hurwitz gegeben. Insbesondere ihre akademische Laufbahn und ihr Einfluss auf die Entwicklung der komplexen Kettebruchtheorie werden beleuchtet. Im zweiten zahlentheoretischen Teil werden verschiedene Perspektiven auf Ergebnisse zurDie Arbeit ’Hurwitz’s Complex Continued Fractions - A Historical Approach and Modern Perspectives.’ beschäftigt sich übergreifend mit zwei Teilbereichen der Mathematik: Zahlentheorie und Geschichte der Mathematik. Im ersten Teil wird ein historischer Blick auf das Leben und Wirken der Gebrüder Adolf und Julius Hurwitz gegeben. Insbesondere ihre akademische Laufbahn und ihr Einfluss auf die Entwicklung der komplexen Kettebruchtheorie werden beleuchtet. Im zweiten zahlentheoretischen Teil werden verschiedene Perspektiven auf Ergebnisse zur Approximationsgüte gegben. Hier stehen ein ergodentheoretisches Resultat, ein komplexes Analogon zur Döblin-Lenstra-Vermutung, im Vordergrund, sowie eine Verallgemeinerung auf Gitter. Aufbauend auf Methoden aus der ’Geometrie der Zahlen’ von Hermann Minkowski wird eine obere Approximationsschranke für allgemeine Gitter gegeben.…