Rencontres de théorie analytique des nombres
# Explicit (and improved) results on the structure of sumsets

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Salle 201, IHP, Paris
### Salle 201, IHP, Paris

Description

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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Régis de la Bretèche

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