Strong bounds for three-term progressions

ACO Colloquium
Friday, June 30, 2023 - 11:00am for 1 hour (actually 50 minutes)
Klaus 2100
Raghu Meka – UCLA –
Santosh Vempala

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 this is the case when C is roughly (log log N). Behrend in 1946 showed that C can be at most exp(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 showing that C can be as big as exp((log N)^0.08), thus getting closer to Behrend's construction. Based on joint work with Zander Kelley