Explicit (and improved) results on the structure of sumsets

Let $A$ be a finite set of integer lattice points in $d$ dimensions, with $NA$ being the set of all sums of $N$ elements from $A$. In 1992 Khovanskii proved the remarkable result that there is a polynomial $P(N)$, depending only on $A$, such that the
size of $NA$ equals $P(N)$ exactly, once $N$ is sufficiently large. Khovanskii's theorem shows that the sumset $NA$ enjoys a
certain size 'stability' property, and there is another related stability property pertaining to the structure of $NA$. But what
does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain
joint work with A. Granville and G. Shakan which proves the first explicit bounds for all sets A. I will also discuss
current work with Granville, which gives an optimal bound 'up to logarithmic factors'.