Strong Bounds for 3-Progressions: In-Depth

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 ≈ (log ⁡ log ⁡ N), while Behrend in 1946 showed that C can be at most 2 Ω(√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 greater than 0.

This talk will describe a new work showing that the same holds when C ≈ 2(log ⁡ N)0.09, thus getting closer to Behrend's construction.