On the expressive power of mod-$p$ linear forms on the Boolean cube

TL;DR

This paper investigates the correlation properties of mod-$p$ linear forms on dense subsets of the Boolean cube, revealing that large collections exhibit almost positive correlation or significant overlap in their distributions.

Contribution

It establishes new bounds on the correlation and overlap of mod-$p$ linear forms across large families of dense Boolean cube subsets.

Findings

Superpolynomial size families have almost positively correlated distributions.

Large enough families have distributions with overlap bounded below by a positive constant.

Results depend only on the size of the family and the prime p.

Abstract

Let $(\mathcal{A}_i)_{i \in [s]}$ be a sequence of dense subsets of the Boolean cube $\{0,1\}^n$ and let $p$ be a prime. We show that if $s$ is assumed to be superpolynomial in $n$ then we can find distinct $i,j$ such that the two distributions of every mod-$p$ linear form on $\mathcal{A}_i$ and $\mathcal{A}_j$ are almost positively correlated. We also prove that if $s$ is merely assumed to be sufficiently large independently of $n$ then we may require the two distributions to have overlap bounded below by a positive quantity depending on $p$ only.