On the Green-Tao theorem for sparse sets

We establish the following quantitative form of the Green–Tao theorem: if a set $\mathcal{A}$ of relative density $\delta$ within the primes up to $N$ contains no nontrivial arithmetic progressions of length $k\ge 4$, then $\delta \ll \exp (-(\log \log \log N{)}^{c_k})$ 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.