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arXiv · 2603.09281

On the Green-Tao theorem for sparse sets

Joni Teräväinen·Mengdi Wang
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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.

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Joni Teräväinen, Mengdi Wang. 2026-03-10. On the Green-Tao theorem for sparse sets. https://arxiv.org/abs/2603.09281

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