Rencontres de théorie analytique des nombres

Explicit (and improved) results on the structure of sumsets

by Aled Walker (King's College, Londres, Angleterre)

Salle 201, IHP, Paris

Salle 201, IHP, Paris


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'.

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