A four vertex theorem for frieze patterns?

TL;DR

This paper investigates sign change properties in differences of two Coxeter frieze patterns, establishing a four-vertex-like theorem for specific rows and an infinitesimal version, with related expository content.

Contribution

It proves a four-vertex theorem analogue for certain rows of Coxeter frieze patterns and introduces an infinitesimal version of the problem.

Findings

First and second non-trivial rows must change sign at least four times.

All rows satisfy the sign change property in the infinitesimal case.

Includes expository material on the four vertex theorem and Coxeter's frieze patterns.

Abstract

Given two Coxeter's frieze patterns with the same width and consisting of positive numbers, choose a row and consider the periodic sequence of the differences of the respective entries of the two friezes. We ask for which rows this sequence must change sign at least four times over the period. We prove that this is the case for the first and for the second non-trivial rows, and that this is true, for all rows, for an infinitesimal version of the question. The article also contains expository material on the four vertex theorem and on Coxeter's frieze patterns.