Effective results on the size and structure of sumsets

Let $A\subset {\mathbb{Z}}^d$ be a finite set. It is known that $NA$ has a particular size ($\mathrm{\mid }NA\mathrm{\mid }=P_A(N)$ 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 $\mathrm{\mid }A\mathrm{\mid }=d+2$, results which we improve.