On sums and products of integers

Erdos and Szemerédi conjectured that if $A$ is a set of $k$ positive integers, then there must be at least $k^{2-\varepsilon }$ integers that can be written as the sum or product of two elements of $A$. Erdos and Szemerédi proved that this number must be at least $c k^{1 + \delta }$ for some $\delta > 0$ and $k \geq k_0$. In this paper it is proved that the result holds for $\delta = 1/31$.