Extractors in Paley graphs: A random model

Abstract

A well-known conjecture in analytic number theory states that for every pair of sets X,Y⊂Z/pZ, each of size at least logCp (for some constant C) we have that the number of pairs (x,y)∈X×Y such that x+y is a quadratic residue modulo p differs from 12|X||Y| by o(|X||Y|). We address the probabilistic analogue of this question, that is for every fixed δ>0, given a finite group G and A⊂G a random subset of density 12, we prove that with high probability for all subsets |X|,|Y|⩾log2+δ|G|, the number of pairs (x,y)∈X×Y such that xy∈A differs from 12|X||Y| by o(|X||Y|).