A finiteness condition for complex continued fraction algorithms

Proceedings of the American Mathematical Society

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 α {\boldsymbol {\alpha }} -Hurwitz algorithms with parameters α ∈ Q 2 {\boldsymbol {\alpha }}\in \mathbb {Q}^2 . This is equivalent to the existence of certain partitions with finitely many atoms 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 [Conform. Geom. Dyn. 29 (2025), pp. 57–89].

Authors 3

1. Charlene Kalle Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Leiden University, Einsteinweg 55, 2333CC Leiden, The Netherlands

2. Fanni M. Sélley Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Leiden University, Einsteinweg 55, 2333CC Leiden, The Netherlands

3. Jörg Μ. Thuswaldner

   Montanuniversität Leoben

   Affiliation as printed

   Lehrstuhl für Mathematik und Statistik, Montanuniversität Leoben, Franz Josef Straße 18, A-8700 Leoben, Austria