Effective Results on the Size and Structure of Sumsets

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 \(\vert A\vert = d+2\) Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.