# Caltech/UCLA Joint Analysis Seminar

How much additive structure can we guarantee in sets of integers, knowing only their density? The study of which density thresholds are sufficient to guarantee the existence of various kinds of additive structures is an old and fascinating subject with connections to analytic number theory, additive combinatorics, and harmonic analysis.

In this talk we will discuss recent progress on perhaps the most well-known of these thresholds: how large do we need a set of integers to be to guarantee the existence of a three-term arithmetic progression? In recent joint work with Olof Sisask we broke through the logarithmic density barrier for this problem, establishing in particular that if a set is dense enough such that the sum of reciprocals diverges, then it must contain a three-term arithmetic progression, establishing the first case of an infamous conjecture of Erdos.

We will give an introduction to this problem and sketch some of the recent ideas that have made this progress possible. We will pay particular attention to the ways we exploit 'spectral structure' - understanding combinatorially sets of large Fourier coefficients, which we hope will have further applications in number theory and harmonic analysis.