On sums and products of integers

Erdös and Szemerédi proved that if $A$ is a set of $k$ positive integers, then there must be at least $ck^{1+\delta}$ integers that can be written as the sum or product of two elements of $A$, where $c$ is a constant and $\delta>0$. Nathanson proved that the result holds for $\delta=\frac 1{31}$. In this paper it is proved that the result holds for $\delta=\frac 15$ and $c=\frac 1{20}$.