Joint TCS - Combinatorics Seminar

Speaker:  Raghu Meka  (UCLA)

Title:  Strong bounds for 3-progressions

Abstract:

Suppose you have a set S of integers from \{1,2,...,N \ that contains at least N / C elements. Then for large enough N , must S contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?

In 1953, Roth showed that this is indeed the case when C > \Omega(\log \log N), while Behrend in 1946 showed that C can be at most 2^{O(\sqrt{\log N})}. Since then, the problem has been a cornerstone of the area of additive combinatorics. Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on C to C = (\log N)^{1+c}, for some constant c > 0.

This talk will describe a new work that C >2^{\Omega((\log N)^{0.09), thus getting closer to Behrend's construction. Based on joint work with Zander Kelley.

The first hour of the talk will be self-contained and describe the main ideas of the proof. The second hour will be a deeper follow-up of some elements of the proof.