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 there is some such that for every and with , there is some nonzero such that contains at least three-term arithmetic progressions with common difference .
We prove that the minimum in Green's theorem is an exponential tower of twos of height on the order of . 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.
Cite this article
Jacob Fox, Huy Tuan Pham, Yufei Zhao, Tower-type bounds for Roth’s theorem with popular differences. J. Eur. Math. Soc. 25 (2023), no. 10, pp. 3795–3831DOI 10.4171/JEMS/1271