Improving Roth's theorem in the primes

TL;DR

This paper improves the density conditions under which subsets of primes necessarily contain three-term arithmetic progressions, advancing the understanding of additive structures in prime sets.

Contribution

It establishes a weaker density threshold involving triple iterated logarithms, extending Green's earlier results on primes containing arithmetic progressions.

Findings

Proves existence of 3-term APs in prime subsets with lower density.

Introduces new techniques to handle sparser prime subsets.

Reduces the density requirement compared to previous bounds.

Abstract

Let A be a subset of the primes. Let \delta_P(N) = \frac{|\{n\in A: n\leq N\}|}{|\{\text{$n$ prime}: n\leq N\}|}. We prove that, if \delta_P(N)\geq C \frac{\log \log \log N}{(\log \log N)^{1/3}} for N\geq N_0, where C and N_0 are absolute constants, then A\cap [1,N] contains a non-trivial three-term arithmetic progression. This improves on B. Green's result, which needs \delta_P(N) \geq C' \sqrt{\frac{\log \log \log \log \log N}{\log \log \log \log N}}.