  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.
