On wild frieze patterns

TL;DR

This paper explores the existence and properties of wild SLk-frieze patterns, demonstrating the presence of both periodic and non-periodic patterns with positive integer entries, including one with infinitely many distinct entries.

Contribution

It establishes the existence of various types of wild SLk-frieze patterns with positive integer entries, including periodic, non-periodic, and infinitely diverse patterns.

Findings

Existence of periodic wild SLk-frieze patterns with positive integers

Existence of non-periodic wild SLk-frieze patterns with positive integers

An SL3-frieze pattern with infinitely many distinct positive integer entries

Abstract

In this note, among other things, we show: There are periodic wild SLk-frieze patterns whose entries are positive integers. There are non-periodic SLk-frieze patterns whose entries are positive integers. There is an SL3-frieze pattern whose entries are positive integers and with infinitely many different entries.