Geometric structure of sumsets

TL;DR

This paper investigates the relationship between sumsets and lattice points in dilated convex hulls of finite lattice point sets, revealing that sumsets approximate the central lattice points of polytopes.

Contribution

It demonstrates that sumsets fill all central lattice points in convex hulls, providing a new perspective on their geometric structure and approximation properties.

Findings

Sumsets occupy all central lattice points in convex hulls.

Sumsets and lattice points in dilated convex hulls grow polynomially.

Sumsets serve as an approximation to lattice points in polytopes.

Abstract

Given a finite set of lattice points, we compare its sumsets and lattice points in its dilated convex hulls. Both of these are known to grow as polynomials. Generally, the former are subsets of the latter. In this paper, we will see that sumsets occupy all the central lattice points in convex hulls, giving us a kind of approximation to lattice points in polytopes.