Strong Bounds for 3-Progressions

Zander Kelley

We show that for some constant β>0, any subset A of integers {1,…,N} of size at least 2^{−O((log N)^β)} N contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least N/(log N)^1+c for a constant c>0.

Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to apply some analogous variants of these same techniques, suitably adapted for the more complicated setting of integers.

This is joint work with Raghu Meka