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Mathematics > Number Theory

arXiv:1805.08007 (math)
[Submitted on 21 May 2018 (v1), last revised 19 Mar 2020 (this version, v4)]

Title:Purely Periodic and Transcendental Complex Continued Fractions

Authors:Gerardo Gonzalez Robert
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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.
Comments: Minor typos were fixed and color images are now black and white. The paper will appear in Acta Arithmetica
Subjects: Number Theory (math.NT)
MSC classes: 11K60, 11J70
Cite as: arXiv:1805.08007 [math.NT]
  (or arXiv:1805.08007v4 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1805.08007
Related DOI: https://doi.org/10.4064/aa180502-17-6

Submission history

From: Gerardo González Robert Mr. [view email]
[v1] Mon, 21 May 2018 12:07:50 UTC (160 KB)
[v2] Thu, 31 Jan 2019 20:58:08 UTC (156 KB)
[v3] Mon, 17 Jun 2019 03:45:54 UTC (156 KB)
[v4] Thu, 19 Mar 2020 22:49:31 UTC (156 KB)
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