# Tower-type bounds for Roth’s theorem with popular differences

### Jacob Fox

Stanford University, USA### Huy Tuan Pham

Stanford University, USA### Yufei Zhao

Massachusetts Institute of Technology, Cambridge, USA

## 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 $ϵ>0$ there is some $N_{0}(ϵ)$ such that for every $N≥N_{0}(ϵ)$ and $A⊂[N]$ with $∣A∣=αN$, there is some nonzero $d$ such that $A$ contains at least $(α_{3}−ϵ)N$ three-term arithmetic progressions with common difference $d$.

We prove that the minimum $N_{0}(ϵ)$ in Green's theorem is an exponential tower of twos of height on the order of $g(1/ϵ)$. 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. 3795–3831

DOI 10.4171/JEMS/1271