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 $M$.
Categories: math.NT math.CO