Good's Theorem for Hurwitz Continued Fractions

Abstract

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

Some typos were fixed. The paper has been published by the International Journal of Number Theory. The published version has a detailed proof of Lemma 2.1