Effective results on the size and structure of sumsets

TL;DR

This paper provides the first effective upper bounds for the threshold size at which sumsets of finite lattice sets exhibit predictable size and structure, extending known results beyond special cases.

Contribution

It introduces explicit upper bounds for the sumset threshold for arbitrary finite sets in integer lattices, improving upon previous special-case results.

Findings

Established effective bounds for the sumset threshold in arbitrary dimensions.

Extended known results from special cases to general finite sets.

Improved upon previous bounds for specific configurations like simplices.

Abstract

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional sets), once $N$ is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary $A$. Such explicit results were only previously known in the special cases when $d=1$, when the convex hull of $A$ is a simplex or when $\vert A\vert = d+2$, results which we improve.