Olof Sisask: On the Kelley-Meka bounds for Roth's theorem

Abstract.

In the 1950s, Klaus Roth used Fourier analysis in a very elegant way to prove a conjecture of Erdos and Turan: he showed that any subset of {1,2,...,N} with more than about N/loglog N elements must contain a non-trivial three-term arithmetic progression. Roth's proof has become very influential in additive number theory and has inspired far-reaching generalisations. In February of this year, Zander Kelley and Raghu Meka released a sensational preprint proving a very strong bound for Roth's theorem. Their argument is also analytic, but of quite a different flavour to Roth's. In this talk I will attempt to give an idea of some of its ingredients, and describe joint work with Thomas Bloom that answers some of the questions raised in Kelley and Meka's preprint.