Tower-type bounds for Roth's theorem with popular differences

Abstract

Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $\epsilon > 0$ there is some $N_0(\epsilon)$ such that for every $N \ge N_0(\epsilon)$ and $A \subset [N]$ with $|A| = \alpha N$, there is some nonzero $d$ such that $A$ contains at least $(\alpha^3 - \epsilon) N$ three-term arithmetic progressions with common difference $d$. We prove that the minimum $N_0(\epsilon)$ in Green's theorem is an exponential tower of 2s of height on the order of $\log(1/\epsilon)$. Both the lower and upper bounds are new. It shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.