Strong Bounds for 3-Progressions

We show that for some constant $\beta\gt0$, any subset A of integers $\{1, \ldots, N\}$ of size at least $2^{-O\left((\log N)^{\beta}\right)} \cdot N$ 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 $c\gt0$.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.