An arithmetic transference proof of a relative Szemer\'edi theorem

TL;DR

This paper presents a more direct proof of a relative Szemerédi theorem using arithmetic transference, improving bounds and simplifying the approach compared to previous hypergraph-based methods.

Contribution

It offers an alternative proof that directly transfers Szemerédi's theorem via a transference principle, avoiding hypergraph removal and yielding better quantitative bounds.

Findings

Provides a more direct proof of the relative Szemerédi theorem.

Achieves improved quantitative bounds over previous methods.

Simplifies the proof by using discrepancy norms instead of Gowers uniformity norms.

Abstract

Recently Conlon, Fox, and the author gave a new proof of a relative Szemer\'edi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemer\'edi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition. This note contains an alternative proof of the new relative Szemer\'edi theorem, where we directly transfer Szemer\'edi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds. The proof has three main ingredients: (1) a transference principle/dense model theorem of Green-Tao and Tao-Ziegler (with simplified proofs given later by Gowers, and independently, Reingold-Trevisan-Tulsiani-Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works), (2) a counting lemma established by Conlon, Fox, and the author, and (3) Szemer\'edi's theorem as a black box.