Sums of sets of lattice points and unimodular coverings of polytopes

TL;DR

This paper investigates the relationship between lattice points in polytopes and their sumsets, proving that for certain unions of unimodular simplices, all lattice points in the sumset are sums of lattice points in the original polytope.

Contribution

It establishes that if a lattice polytope is a union of unimodular simplices, then every lattice point in its h-fold sumset can be expressed as a sum of h lattice points from the polytope.

Findings

Lattice points in the h-fold sumset correspond to sums of lattice points in the polytope under certain conditions.

Unimodular simplex unions ensure the sumset property holds.

The result generalizes the understanding of sumsets in lattice polytopes.

Abstract

If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in $\mathbf{R}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the $h$-fold sumset $hP$ is not necessarily the sum of $h$ lattice points in $P$. It is proved that if the polytope $P$ is a union of unimodular simplices, then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.