On Sums of Sets of Primes with Positive Relative Density

TL;DR

This paper proves that subsets of primes with positive relative density have sumsets with positive upper density, extending understanding of additive properties of prime subsets using advanced combinatorial and number theoretic techniques.

Contribution

It introduces a novel combination of Green and Green-Tao's methods with multiplicative subgroup sum results to analyze sumsets of prime subsets.

Findings

Sumsets of prime subsets with positive density have positive upper density.

The result extends additive combinatorics techniques to prime subsets.

Method combines Green-Tao techniques with multiplicative subgroup analysis.

Abstract

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.