On the Green-Tao theorem for sparse sets

Abstract

We establish the following quantitative form of the Green--Tao theorem: if a set $\mathcal{A}$ of relative density $\hat{I}´$ within the primes up to $N$ contains no nontrivial arithmetic progressions of length $k\ge 4$, then $\hat{I}´\ll \exp (-(\log \log \log N{)}^{c_k})$ for some $c_k>0$. This improves on previous work of Rimanić and Wolf. The main new ingredients in the proof are a version of the Leng--Sah--Sawhney quasipolynomial inverse theorem for unbounded functions and a dense model theorem with quasipolynomial dependencies, which may be of independent interest.

46 pages