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On the Green-Tao theorem for sparse sets

Joni Teräväinen, Mengdi Wang

Abstract

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

On the Green-Tao theorem for sparse sets

Abstract

We establish the following quantitative form of the Green--Tao theorem: if a set of relative density within the primes up to contains no nontrivial arithmetic progressions of length , then for some . 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.
Paper Structure (12 sections, 22 theorems, 265 equations)

This paper contains 12 sections, 22 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Organisation of the paper
  4. Notation
  5. Transferred inverse theorem for Gowers uniformity norms
  6. Transferred regularity lemmas (dense model)
  7. Transferred regularity lemma I
  8. Transferred regularity lemma II
  9. Correlation of the von Mangoldt function with nilsequences
  10. Generalised von Neumann theorem
  11. Transference principle and the sparse Green--Tao theorem
  12. Type I and Type II Estimates for Nilsequences

Key Result

Theorem 1.1

Let $k\geqslant 4$ be a natural number, and let $N\geqslant 100$. Let $\mathcal{A}\subseteq[N]\cap \mathbb P$ be a subset of primes. If $\mathcal{A}$ does not contain any nontrivial $k$-term arithmetic progression then for some $c_k>0$ we have

Theorems & Definitions (56)

  • Theorem 1.1: Density bound
  • Lemma 1.2: Interval Gowers--Cauchy--Schwarz
  • proof
  • Definition 2.1: Filtered nilmanifold
  • Definition 2.2: Nilsequences
  • Theorem 2.3: Quasipolynomial inverse theorem
  • proof
  • Lemma 2.4: Product of nilsequences
  • proof
  • Theorem 2.5: Transferred inverse theorem---quasipolynomial bounds
  • ...and 46 more