Aled Walker

Aled Walker, King's College, Londres, Angleterre


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'.