Explicit (and improved) results on the structure of sumsets

Given a finite set $A$ of integer lattice points in $d$ dimensions, let $NA$ denote the $N$-fold iterated sumset (i.e. the set comprising all sums of $N$ elements from $A$). In 1992 Khovanskii observed that there is a fixed polynomial $P(N)$, depending on $A$, such that the size of the sumset $NA$ equals $P(N)$ exactly (once $N$ is sufficiently large, that is). In addition to this 'size stability', there is a related 'structural stability' property for the sumset $NA$, which Granville and Shakan recently showed also holds for sufficiently large $N$. But what does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain joint work with Granville and Shakan which proves the first explicit bounds for all sets $A$. I will also discuss current work with Granville, which gives a tight bound 'up to logarithmic factors' for one of these properties.