Bourbaki - Juin 2022

Collection Bourbaki - Juin 2022

Date(s) 18/06/2022 - 18/06/2022
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[1196] Recent progress on bounds for sets with no three terms in arithmetic progression

By Sarah Peluse

Also appears in collection : Distinguished women in mathematics

A famous conjecture of Erdős states that if $S$ is a subset of the positive integers and the sum of the reciprocals of elements of $S$ diverges, then $S$ contains arbitrarily long arithmetic progressions. If one could prove, for each positive integer $k$, sufficiently good bounds for the size of the largest subset of the first $N$ integers lacking $k$-term arithmetic progressions, then Erdős’s conjecture would follow. There is thus great interest in the problem of proving the strongest possible bounds for sets lacking arithmetic progressions of a fixed length. In this talk, I will survey the recent advances of Bloom–Sisask on this problem for length three progressions and of Croot–Lev–Pach and Ellenberg–Gijswijt on the analogous problem in $F^n_3$ (the "cap set problem"). These two advances rely on very different techniques —Fourier analytic methods and a version of the polynomial method, respectively— and I will give an overview of both.

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  • Séminaire Bourbaki, 74ème année (2021-2022), n°1196, juin 2022 PDF

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