Séminaire Bourbaki
# Sarah Peluse — Recent progress on bounds for sets with no three terms in arithmetic progression

→
Europe/Paris

Hermite (IHP)
### Hermite

#### IHP

Description

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.