Noise Correlation Bounds for Uniform Low Degree Functions

TL;DR

This paper establishes correlation bounds for low-degree, uniform functions under pairwise independent distributions, with implications for complexity theory and additive combinatorics.

Contribution

It provides new bounds for correlations of low-degree, uniform functions under pairwise independence, extending hypercontractivity techniques to multilinear cases.

Findings

Correlation is close to zero under specified conditions.

Bounds extend previous results for low influence functions.

Connections to additive combinatorics and Gowers norm.

Abstract

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by $\delta$ are called $\delta$-{\em uniform}. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics. In our main result we show that $\E[f_1(X_1^1,...,X_1^n) ... f_k(X_k^1,...,X_k^n)]$ is close to 0 under the following assumptions: 1. The vectors $\{(X_1^j,...,X_k^j) : 1 \leq j \leq n\}$ are i.i.d, and for each $j$ the vector $(X_1^j,...,X_k^j)$ has a pairwise independent distribution. 2. The functions $f_i$ are uniform. 3. The functions $f_i$ are of low degree. We compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.