Effective Results on the Size and Structure of Sumsets

Let A subset of Z(d) be a finite set. It is known that N A has a particular size (vertical bar N A vertical bar| = P-A(N) for some P-A(X) is an element of 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 vertical bar A vertical bar = d + 2 Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.