Extractors in Paley graphs: a random model

A well-known conjecture in analytic number theory states that for every pair of sets $X,Y\subset \mathbb{Z}\mathrm{/}p\mathbb{Z}$, each of size at least ${\log }^{C}p$ (for some constant $C$) we have that the number of pairs $(x,y)\in X\times Y$ such that $x+y$ is a quadratic residue modulo $p$ differs from $\frac{1}{2}\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }$ by $o\left(\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }\right)$. We address the probabilistic analogue of this question, that is for every fixed $\delta >0$, given a finite group $G$ and $A\subset G$ a random subset of density $\frac{1}{2}$, we prove that with high probability for all subsets $\mathrm{\mid }X\mathrm{\mid },\mathrm{\mid }Y\mathrm{\mid }\ge {\log }^{2+\delta }\mathrm{\mid }G\mathrm{\mid }$, the number of pairs $(x,y)\in X\times Y$ such that $xy\in A$ differs from $\frac{1}{2}\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }$ by $o\left(\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }\right)$.