A relative Szemer\'edi theorem

TL;DR

This paper presents a simplified and strengthened proof of the relative Szemerédi theorem, requiring weaker pseudorandomness conditions, and extends the regularity method to sparse hypergraphs to find long arithmetic progressions in pseudorandom sets.

Contribution

The authors provide a simplified proof of a stronger relative Szemerédi theorem, applicable to sparser pseudorandom subsets, and extend the regularity method to sparse hypergraphs.

Findings

Proved a strengthened relative Szemerédi theorem under weaker pseudorandomness conditions.

Extended the regularity method to sparse pseudorandom hypergraphs.

Derived a relative hypergraph removal lemma applicable to sparse sets.

Abstract

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\'edi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemer\'edi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemer\'edi theorem for $k$-term arithmetic progressions in pseudorandom subsets of $\mathbb{Z}_N$ of density $N^{-c_k}$. The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemer\'edi theorem.