New irrational polygons with Ehrhart-theoretic period collapse

TL;DR

This paper introduces new irrational polygons exhibiting Ehrhart-theoretic period collapse, expanding the class of polygons with polynomial lattice-point counting functions and suggesting potential applications in symplectic geometry.

Contribution

It generalizes previous irrational triangle examples to polygons with more sides using a simple cut-and-paste method, revealing new instances of Ehrhart polynomial behavior.

Findings

Most constructed polygons have Ehrhart functions that are polynomials.

The method allows building polygons with more sides exhibiting period collapse.

Potential applications in symplectic embedding theory.

Abstract

In a recent paper, Cristofaro-Gardiner--Li--Stanley [CGLS15] constructed examples of irrational triangles whose Ehrhart functions (i.e. lattice-point count) are polynomials when restricted to positive integer dilation factors. This is very surprising because the Ehrhart functions of rational polygons are usually only quasi-polynomials. We demonstrate that most of their triangles can also be obtained by a simple cut-and-paste procedure that allows us to build new examples with more sides. Our examples might potentially have applications in the theory of symplectic embeddings.