On sums of finite subsets of the primes

Abstract

Let $A\subset [1,x]$ be a non-empty set of primes with $|A|= \alpha x(\log x)^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $|A+A|\geq c_1(\log x)(\log \log 3\alpha^{-1})^{-1}|A|$ if $\alpha \geq c_2(\log x)^{-1/2}\log \log x$ and otherwise $|A+A|\geq c_1(\log x) (\log 2\alpha^{-1})^{-1}|A|$.