Extractors in Paley graphs: a random model

TL;DR

This paper investigates a probabilistic analogue of a conjecture in analytic number theory, demonstrating that for large enough subsets in a finite group, the distribution of product pairs in a random subset approximates uniformity with high probability.

Contribution

It proves that in a finite group, random subsets of density 1/2 exhibit near-uniform pairwise distribution for all sufficiently large subsets, extending a classical conjecture to a probabilistic setting.

Findings

High probability of near-uniform distribution in random subsets

Valid for all subsets larger than logarithmic powers of group size

Supports probabilistic models for additive number theory conjectures

Abstract

A well-known conjecture in analytic number theory states that for every pair of sets $X,Y\subset\mathbb{Z}/p\mathbb{Z}$, 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 $\frac12|X||Y|$ by $o\left(|X||Y|\right)$. 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 $\frac12$, we prove that with high probability for all subsets $|X|,|Y|\geq \log ^{2+\delta} |G|$, the number of pairs $(x,y)\in X\times Y$ such that $xy\in A$ differs from $\frac12|X||Y|$ by $o\left(|X||Y|\right)$.