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