Ehrhart spectra of large subsets of Zʳ

This paper introduces and studies the Ehrhart spectrum of a set E ⊆ Zʳ, 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 ⊆ Zʳ with positive upper Banach density, there is some n ∈ Zʳ such that the Ehrhart spectrum of n Zʳ is contained in the Ehrhart spectrum of E, generalizing an earlier result by the first and third author for the volume spectrum of E.