Rencontres de théorie analytique des nombres

Explicit (and improved) results on the structure of sumsets

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

Salle Grisvard, IHP, Paris

Salle Grisvard, IHP, Paris


Given a finite set $A$ of integer lattice points in $d$ dimensions, let $NA$ denote the $N$-fold iterated sumset (i.e. the set comprising all sums of $N$ elements from $A$). In 1992 Khovanskii observed that there is a fixed polynomial $P(N)$, depending on $A$, such that the size of the sumset $NA$ equals $P(N)$ exactly (once $N$ is sufficiently large, that is). In addition to this 'size stability', there is a related 'structural stability' property for the sumset $NA$, which Granville and Shakan recently showed also holds for sufficiently large $N$. But what does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain joint work with Granville and Shakan which proves the first explicit bounds for all sets $A$. I will also discuss current work with Granville, which gives a tight bound 'up to logarithmic factors' for one of these properties.


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