Good's Theorem for Hurwitz Continued Fractions

TL;DR

This paper extends Good's Theorem from real to complex numbers, showing that the set of complex numbers with Hurwitz continued fractions diverging in partial quotients has Hausdorff dimension 1.

Contribution

It establishes an analogue of Good's Theorem for Hurwitz continued fractions in the complex plane, revealing the Hausdorff dimension of a divergence set.

Findings

Hausdorff dimension of divergence set is 1 in the complex plane

Analogue of Good's Theorem holds for Hurwitz continued fractions

Divergence in partial quotients corresponds to full dimension set

Abstract

Good's Theorem for regular continued fraction states that the set of real numbers $[a_0;a_1,a_2,\ldots]$ such that $\displaystyle\lim_{n\to\infty} a_n=\infty$ has Hausdorff dimension $\tfrac{1}{2}$. We show an analogous result for the complex plane and Hurwitz Continued Fractions. The set of complex numbers whose Hurwitz Continued fraction $[a_0;a_1,a_2,\ldots]$ satisfies $\displaystyle\lim_{n\to\infty} |a_n|=\infty$ has Hausdorff dimension $1$, half of the ambient space's dimension.