Ehrhart spectra of large subsets of ${\mathbb{Z}}^r$

This paper introduces and studies the Ehrhart spectrum of a set $E\subseteq {\mathbb{Z}}^r$, defined as the set of all Ehrhart polynomials of simplices with vertices in $E$, generalizing the notion of volume spectrum. We show that for any $E\subseteq {\mathbb{Z}}^r$ with positive upper Banach density, there is some $n\in {\mathbb{Z}}^r$ such that the Ehrhart spectrum of $n{\mathbb{Z}}^r$ is contained in the Ehrhart spectrum of $E$, generalizing an earlier result by the first and third author for the volume spectrum of $E$.