Pick's theorem and sums of lattice points

TL;DR

This paper uses Pick's theorem to prove a property of lattice points in scaled sums of lattice polygons, showing that all lattice points in the h-fold sumset are sums of h lattice points in the original polygon.

Contribution

It provides a novel proof leveraging Pick's theorem to characterize lattice points in the sumsets of lattice polygons.

Findings

All lattice points in hP are sums of h lattice points in P.

The proof connects geometric properties with additive combinatorics.

The result applies to convex lattice polygons in the plane.

Abstract

Pick's theorem is used to prove that if $P$ is a lattice polygon (that is, the convex hull of a finite set of lattice points in the plane), then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.