A finiteness condition for complex continued fraction algorithms

TL;DR

This paper establishes that complex α-Hurwitz continued fraction algorithms with rational parameters satisfy a key finiteness condition, enabling easier characterization of their representations and advancing understanding of their Diophantine properties.

Contribution

It proves the finite range condition for a class of complex continued fraction algorithms with rational parameters, addressing a recent open question.

Findings

Finite range condition holds for complex α-Hurwitz algorithms with rational parameters.

Existence of finite partitions related to these algorithms is established.

Results contribute to understanding Diophantine properties of complex continued fractions.

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.