Georgia Institute of Technology College of Sciences

Suppose you have a set $A$ of integers from $\{1, 2, …, N\}$ that contains at least $N / C$ elements.

Then for large enough $N$, must $A$ 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 $2^{\sqrt{\log N}}$ by giving an explicit construction of a large set with no 3-term progressions.

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 which shows that the same holds when $C \approx 2^{(\log N)^{1/12}}$, thus getting closer to Behrend's construction.

Based on a joint work with Raghu Meka.