Metrical properties of Hurwitz Continued Fractions

TL;DR

The paper extends the theory of Hurwitz continued fractions to the complex plane, developing both the geometric structure of the HCF expansions and a metrical theory that mirrors real-case results. It proves that the space of valid partial quotient sequences is not closed and computes the Hausdorff dimension of sets defined by fast growth of partial quotients, unifying upper and lower bounds via two geometric constructions and a Cantor-type mass distribution argument. The main contributions include a complete dimension formula: dim_H$(E_ ext{∞}(\Phi))$ equals 2 when $B=1$, equals $s_B$ for $1<B<\infty$, and equals $2/(1+b)$ for $B=\infty$ with $\log b = \liminf_{n\to\infty} \log\log\Phi(n)/n$, together with a detailed symbolic-geometric framework for Hurwitz CF cylinders and their interactions. This advances complex Diophantine approximation by Gaussian integers and provides tools for understanding extremal approximation sets in the complex plane.

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 $Φ:\mathbb{N}\to \mathbb{R}_{>0}$ 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 \[E(Φ) := \left\{ z\in \mathbb C: |a_n(z)|\geq Φ(n) \text{ for infinitely many }n\in\mathbb{N} \right\}. \] This study is a complex analog of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319--1339].

Figures (12)

• Figure 1: The set $\iota[\mathfrak{F}]$.
• Figure 2: Cylinders of level $1$.
• Figure 3: The set $\iota[\mathfrak{F}_1(-2)]$.
• Figure 4: Closed prototype sets
• Figure 5: Possible configurations

Theorems & Definitions (132)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4: Łuczak, Luc1997
• Theorem 1.5
• Remark 3.1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3: Khi1997
• Definition 5.1