Improved stability for the size and structure of sumsets

Let $A\subset {\mathbb{Z}}^d$ be a finite set. It is known that the sumset $NA$ has predictable 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 some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once $N$ is larger than some threshold. In previous work, joint with Shakan, the first and third named authors established the first effective bounds for both of these thresholds for an arbitrary set $A$. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known.