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 $A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a bound given by Balog, Rivat and S\'ark\"ozy.