Primes in Sumsets

We obtain an upper bound for the number of pairs $(a,b)\in A\times B$ such that $a+b$ is a prime number, where $A,B\subseteq \{1,\mathrm{.}\mathrm{.}\mathrm{.},N\}$ with $\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }\gg \frac{N^2}{(\log N{)}^2}$, $N\ge 1$ an integer. This improves on a bound given by Balog, Rivat and S'ark"ozy.