Improved stability for the size and structure of sumsets

Let A Zᵈ be a finite set. It is known that the sumset NA has predictable size (NA = PA (N) for some PA (X) QX) 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.