Roth's theorem in the primes

TL;DR

This paper proves that any subset of primes with positive density contains a 3-term arithmetic progression, utilizing a new proof of the Hardy-Littlewood majorant property for primes and a restriction theorem analogy.

Contribution

It introduces a novel proof of the Hardy-Littlewood majorant property for primes and establishes a restriction theorem analogy, advancing understanding of prime patterns.

Findings

Any positive density subset of primes contains a 3-term arithmetic progression

New proof of Hardy-Littlewood majorant property for primes

Establishment of a restriction theorem analogy for primes

Abstract

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes.