Purely Periodic and Transcendental Complex Continued Fractions

TL;DR

This paper investigates the properties of Hurwitz continued fractions for complex numbers, providing conditions for pure periodicity, characterizing badly approximable numbers, and exploring transcendence criteria.

Contribution

It offers new necessary and sufficient conditions for pure periodicity of Hurwitz continued fractions and extends transcendence results to complex continued fractions.

Findings

Conditions for pure periodicity of HCF are established.

Badly approximable complex numbers are characterized via HCF.

A complex analogue of Bugeaud's transcendence theorem is proved.

Abstract

Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$ are precisely those with periodic HCF expansions. In this paper, we give some necessary as well as some sufficient conditions for pure periodicity of HCF. Then, we characterize badly approximable complex numbers in terms of HCF. Finally, we prove a slightly weaker complex analogue of a theorem by Y. Bugeaud on the transcendence of certain continued fractions.