Purely periodic and transcendental complex continued fractions
Volume 194 / 2020
Acta Arithmetica 194 (2020), 241-265
MSC: Primary 11K60; Secondary 11J70.
DOI: 10.4064/aa180502-17-6
Published online: 12 March 2020
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 (2003) on the transcendence of certain continued fractions.
