Frieze patterns over integers and other subsets of the complex numbers

TL;DR

This paper explores frieze patterns over complex subsets, introduces new transformations for quiddity cycles, and provides a combinatorial model for integer frieze patterns, linking to cluster algebras and addressing finiteness questions.

Contribution

It introduces a general combinatorial model for tame frieze patterns over integers and extends the understanding of their finiteness properties over various complex subsets.

Findings

New transformations for quiddity cycles

A combinatorial model for integer frieze patterns

Finiteness results for frieze patterns over discrete subsets

Abstract

We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway-Coxeter theory. This model is thus also a model for the set of specializations of cluster algebras of Dynkin type $A$ in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset $R$ of the complex numbers. Under certain conditions on $R$, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if $R$ is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over $R$. Using this, we disprove a conjecture of Fontaine, by showing that for a complex $d$-th root of unity $\zeta_d$ there are only finitely many non-zero frieze patterns for a given height over $R=\mathbb{Z}[\zeta_d]$ if and only if $d\in \{1,2,3,4,6\}$.