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 $|A+A|\ge c_1(\log x)(\log \log 3{\alpha }^{-1}{)}^{-1}|A|$ if $\alpha \ge c_2(\log x{)}^{-1/2}\log \log x$ and otherwise $|A+A|\ge c_1(\log x)(\log 2{\alpha }^{-1}{)}^{-1}|A|$.

10 pages