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arXiv · 2406.18689

A finiteness condition for complex continued fraction algorithms

Charlene Kalle·Fanni M. Sélley·Jörg M. Thuswaldner
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Abstract

It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show that this condition holds for complex $\boldsymbol{\alpha}$-Hurwitz algorithms with parameters $\boldsymbol{\alpha}\in\mathbb{Q}^2$. This is equivalent to the existence of certain finite partitions related to these algorithms and lies at the root of explorations into their Diophantine properties. Our result provides a partial answer to a recent question formulated by Lukyanenko and Vandehey.

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Charlene Kalle, Fanni M. Sélley, Jörg M. Thuswaldner. 2024-06-26. A finiteness condition for complex continued fraction algorithms. https://arxiv.org/abs/2406.18689

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