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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 > 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.

Based on joint work with Zander Kelley.