Metrical properties of Hurwitz Continued Fractions

TL;DR

This paper develops the geometric and metrical theory of Hurwitz continued fractions, analyzing their properties, the structure of valid sequences, and the Hausdorff dimension of sets defined by growth conditions on partial quotients.

Contribution

It introduces a comprehensive metrical framework for Hurwitz continued fractions and determines the Hausdorff dimension of sets characterized by growth conditions on partial quotients.

Findings

The space of valid Hurwitz continued fraction sequences is not closed.

A complete metrical theory for Hurwitz continued fractions is established.

The Hausdorff dimension of sets with partial quotient growth conditions is computed.

Abstract

We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.%, paralleling the classical theory for regular continued fractions in real numbers. Let $\Phi:\mathbb{N}\to \mathbb{R}_{>0}$ be any function. For any complex number $z$ and $n\in\mathbb{N}$, let $a_n(z)$ denote the $n$th partial quotient in the Hurwitz continued fraction of $z$. One of the main results of this paper is the computation of the Hausdorff dimension of the set \[E(\Phi) := \left\{ z\in \mathbb C: |a_n(z)|\geq \Phi(n) \text{ for infinitely many }n\in\mathbb{N} \right\}. \] This study is a complex analog of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319--1339].