On Sums of Sets of Primes with Positive Relative Density

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 $M$.

21 pages, to appear in J. London Math. Soc., short remark added and typos fixed