Szemer\'edi's theorem in the primes

TL;DR

This paper extends Green and Tao's work by establishing results on arbitrarily long arithmetic progressions within the primes, using advanced sieve methods and a quantitative Szemerédi theorem.

Contribution

It combines a quantified relative Szemerédi theorem with Henriot's sieve estimates to prove longer prime progressions.

Findings

Proves existence of arbitrarily long arithmetic progressions in primes.

Improves bounds on relative density for prime progressions.

Utilizes advanced sieve techniques and quantitative combinatorial theorems.

Abstract

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of the primes of relative density tending to zero sufficiently slowly contains a 3-term term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of 3-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemer\'edi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.