A Density Increment Approach to Roth's Theorem in the Primes

TL;DR

This paper proves that any set of primes with divergent harmonic sum contains a 3-term arithmetic progression, using a density increment strategy combined with the transference principle to exploit prime structure.

Contribution

It introduces a novel density increment approach to Roth's theorem specifically tailored for primes, advancing the understanding of arithmetic progressions in prime sets.

Findings

Sets of primes with divergent harmonic sum contain 3-term arithmetic progressions.

The method achieves progress for sets with density at least a polylogarithmic factor.

The approach combines transference principles with density increment techniques.

Abstract

We prove that if $A$ is any set of prime numbers satisfying \[ \sum_{a\in A}\frac{1}{a}=\infty, \] then $A$ must contain a $3$-term arithmetic progression. This is accomplished by combining the transference principle with a density increment argument, exploiting the structure of the primes to obtain a large density increase at each step of the iteration. The argument shows that for any $B>0$, and $N>N_{0}(B)$, if $A$ is a subset of primes contained in $\{1,\dots,N\}$ with relative density $\alpha(N)=(|A|\log N)/N$ at least \[ \alpha(N)\gg_{B}\left(\log\log N\right)^{-B} \] then $A$ contains a $3$-term arithmetic progression.