Metrical properties of Hurwitz Continued Fractions

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].