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 this is the case when C is roughly (log log N). Behrend in 1946 showed that C can be at most exp(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 C can be as big as exp((log N)^0.09), thus getting closer to Behrend’s construction.

Based on joint work with Zander Kelley.

BIO:

Raghu Meka is a professor in Computer Science and is broadly interested in complexity theory, learning theory, algorithm design. More generally, I like probability and combinatorics-related things.