Discrete Equidecomposability and Ehrhart Theory of Polygons

TL;DR

This paper explores the relationship between Ehrhart equivalence and discrete equidecomposability of polygons, constructing examples that distinguish these concepts and examining conditions under which infinite equidecomposability occurs.

Contribution

It introduces the concept of rational finite discrete equidecomposability, constructs polygons that are Ehrhart equivalent but not discretely equidecomposable, and investigates infinite equidecomposability scenarios.

Findings

Ehrhart equivalent polygons may not be discretely equidecomposable.

Removing an edge can enable infinite rational discrete equidecomposability.

Counterexamples to the implication from Ehrhart equivalence to equidecomposability.

Abstract

Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons $P$ and $Q$ are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice $\mathbb{Z} \times \mathbb{Z}$) from $P$ to $Q$. In this paper, we primarily study a particular version of this notion which we call rational finite discrete equidecomposability. We construct triangles that are Ehrhart equivalent but not rationally finitely discretely equidecomposable, thus providing a partial negative answer to a question of Haase--McAllister on whether Ehrhart equivalence implies discrete equidecomposability. Surprisingly, if we delete an edge from each of these triangles, there exists an infinite rational discrete equidecomposability relation between them. Our final section addresses the topic of infinite equidecomposability with concrete examples and a potential setting for further investigation of this phenomenon.