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Complex Continued Fractions. Theoretical Aspects of Hurwitz’s Algorithm

By Gerardo González Robert
PhD Dissertations
November 2018
Abstract:

Inspired by their remarkable properties, mathematicians have sought in several contexts objects similar to regular continued fractions. Hurwitz Continued fractions (HCF), proposed by Adolf Hurwitz in 1887, give a natural analogue in the complex plane. The HCF of a complex number $\zeta$ is a sequence of Gaussian integers obtained from a specific algorithm. Among the similarities between HCF and regular continued fractions we can find the irrationality of a complex number as a necessary and sufficient condition for its HCF to be infinite. In this case, there even are combinatorial conditions on the HCF implying transcendence. The usual characterization of quadratic surds in terms of periodic continued fractions also holds. However, É. Galois' theorem on purely periodic continued fractions is not true unless additional restrictions are imposed.

HCF can be understood through a dynamical perspective. With a suitably defined Gauss map, HCF give rise to an ergodic dynamical system. Although the corresponding shift space has a complicated structure, some classical Hausdorff dimension results still hold. For example, the set of complex numbers with a bounded HCF has full Hausdorff dimension (proved by V. Jarník in 1929 for $\mathbb{R}$) and the Hausdoff dimension of the set of complex numbers with a HCF tending to $\infty$ is half of that of the ambient space's (proved by I.J. Good in 1941 for $\mathbb{R}$). The theory of HCF is far from being completed, and its similarities and discrepancies with their real counterpart stem new problems.

The present thesis is a survey of new and old results on HCF. In the preface, a detailed list of all the novel aspects is given. Additionally, three original research papers are included as appendices. The first two deal with HCF theory. The third one - applying the geometry of regular continued fractions - contains a simple proof of P. Bengoechea's quantitative improvement of Serret's Theorem on equivalent numbers.

Thesis advisor: Simon Kristensen
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