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 twos of height on the order of \log(1/\epsilon) . Both the lower and upper bounds are new. This shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.