On sums of sparse prime subsets

Abstract

For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $\hat{I}´x(\log x{)}^{-1}$ with $\hat{I}´\ge (\log x{)}^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\hat{I}´x(\log \log x{)}^{-1}$.