On the Green-Tao theorem for sparse sets

TL;DR

This paper provides a quantitative bound on the density of subsets of primes lacking long arithmetic progressions, improving previous results by employing advanced inverse theorems and dense model techniques.

Contribution

It introduces a new quasipolynomial inverse theorem and a dense model theorem with quasipolynomial dependencies, enhancing the quantitative understanding of the Green--Tao theorem.

Findings

Established a bound: δ ≪ exp(- (log log log N)^{c_k}) for sets with no k-term APs

Improved previous bounds by Rimanić and Wolf

Developed new inverse and dense model theorems with quasipolynomial complexity

Abstract

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 $k\geq 4$, then $\delta\ll \exp(-(\log \log \log N)^{c_k})$ for some $c_k>0$. This improves on previous work of Rimani\'c 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.