Purely Periodic and Transcendental Complex Continued Fractions
Number Theory
2020-03-23 v4
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 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.
Keywords
Cite
@article{arxiv.1805.08007,
title = {Purely Periodic and Transcendental Complex Continued Fractions},
author = {Gerardo Gonzalez Robert},
journal= {arXiv preprint arXiv:1805.08007},
year = {2020}
}
Comments
Minor typos were fixed and color images are now black and white. The paper will appear in Acta Arithmetica