Period collapse of Markov triangles

TL;DR

This paper extends the phenomenon of period collapse in Ehrhart quasipolynomials from Fibonacci triangles to all Markov triangles, revealing new instances of strong period collapse linked to Markov numbers.

Contribution

It generalizes previous results by using integral affine geometry to demonstrate period collapse for all Markov triangles, including new irrational limits.

Findings

Constructed sequences of rational triangles with Ehrhart quasipolynomial period p

Identified irrational limits with quasipolynomial Ehrhart functions of period p

Demonstrated strong period collapse for all Markov numbers p

Abstract

Cristofaro-Gardiner and Kleinman showed the complete period collapse of the Ehrhart quasipolynomial of Fibonacci triangles and their irrational limits, by studying the Fourier-Dedekind sums involved in the Ehrhart function of right-angled rational triangles. We generalize this result using integral affine geometrical methods to all Markov triangles, as defined by Vianna. In particular, we show new occurrences of strong period collapse, namely by constructing for each Markov number $p$ a two-sided sequence of rational triangles and two irrational limits with quasipolynomial Ehrhart function of period $p$.