Ehrhart spectra of large subsets of $\mathbb{Z}^r$
Michael Björklund, Rickard Cullman, Alexander Fish
TL;DR
This work introduces the Ehrhart spectrum $\mathrm{EhrSpec}(E)$, the collection of Ehrhart polynomials of simplices with vertices in $E\subseteq \mathbb{Z}^r$, as a natural generalization of the volume spectrum and establishes an inclusion principle for large lattice subsets. The authors prove that if $E$ has positive upper Banach density, there exists $n$ with $\mathrm{EhrSpec}(n\mathbb{Z}^r)\subseteq \mathrm{EhrSpec}(E)$, achieved by lifting the problem to a dynamical setting and proving a multi-correlation theorem via random walks on $\operatorname{SL}_r(\mathbb{Z})$ and Furstenberg-type arguments. The core technical contributions are a haystack lemma ensuring that certain positive-measure translation sets are not confined to hyperplanes and a dynamical theorem guaranteeing persistent intersections under a single $\operatorname{SL}_r(\mathbb{Z})$-action, which together yield the spectral inclusion. The results unify valuation theory under $\operatorname{SL}_r(\mathbb{Z})$-invariance and extend previous volume-based findings to the full Ehrhart spectrum, with potential implications for understanding lattice-point enumerations in large dense sets.
Abstract
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 $E$.