The number of rational numbers determined by large sets of integers

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $\mathrm{\mid }A\mathrm{\mid }\ge \alpha X$ and $\mathrm{\mid }B\mathrm{\mid }\ge \beta X$, we show that the number of rational numbers expressible as $a\mathrm{/}b$ with $(a,b)$ in $A\times B$ is $\gg (\alpha \beta {)}^{1+ϵ}XY$ for any $ϵ>0$, where the implied constant depends on $ϵ$ alone. We then construct examples that show that this bound cannot in general be improved to $\gg \alpha \beta XY$. We also resolve the natural generalisation of our problem to arbitrary subsets $C$ of the integer points in $[1,X]\times [1,Y]$. Finally, we apply our results to answer a question of S'ark"ozy concerning the differences of consecutive terms of the product sequence of a given integer sequence.