Bounds for 3-Progressions

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 is roughly loglog(N), while Behrend in 1946 showed that C can be at most 2^(√(log N)) by giving an explicit construction of a large set with no 3-term arithmetic 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 our work which shows that the same holds when C is roughly 2^((log N)^(1/12)), thus getting closer to Behrend’s construction.

Based on joint work with Raghu Meka.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/