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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 $$C \approx (\log \log N)$$, while Behrend in 1946 showed that $$C$$ can be at most $$2^{\Omega(\sqrt{\log N})}$$. 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 the same holds when $$C \approx 2^{(\log N)^{0.09}}$$, thus getting closer to Behrend’s construction.

Based on joint work with Zander Kelley.

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