Transcendence and Normality of Complex Numbers

Felipe García-Ramos, Gerardo González Robert, Mumtaz Hussain

Published 2023-10-30Version 1

Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian rational. However, the resulting space of sequences of Gaussian integers $\Omega$ is not closed. In this paper, we show that $\Omega$ contains a natural subset whose closure $\overline{\mathsf{R}}$ encodes continued fraction expansions of complex non-Gaussian rationals. Furthermore, by means of an algorithm that we develop, we exhibit for each complex non-Gaussian rational $z$ a sequence belonging to $\overline{\mathsf{R}}$ giving a continued fraction expansion of $z$. We also prove that $(\overline{\mathsf{R}}, \sigma)$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. More precisely, we prove that it is a $\Pi_0^3$-complete set. As a second application, we construct a family of complex transcendental numbers with bounded partial quotients.