# Gaussian Bounds for Noise Correlation of Resilient Functions

## TL;DR

This paper extends Gaussian bounds on noise correlation of functions from low influence to functions with vanishing Fourier coefficients, providing new insights into their Fourier structure and correlation properties.

## Contribution

It proves that functions with vanishing Fourier coefficients satisfy Gaussian noise correlation bounds, relaxing the low influence condition and analyzing Fourier support intersections.

## Key findings

- Gaussian bounds hold for functions with vanishing Fourier coefficients.
- Large noisy inner product implies intersecting Fourier supports.
- Extends correlation bounds beyond low influence functions.

## Abstract

Gaussian bounds on noise correlation of functions play an important role in hardness of approximation, in quantitative social choice theory and in testing. The author (2008) obtained sharp gaussian bounds for the expected correlation of $\ell$ low influence functions $f^{(1)},\ldots, f^{(\ell)} : \Omega^n \to [0,1]$, where the inputs to the functions are correlated via the $n$-fold tensor of distribution $\mathcal{P}$ on $\Omega^{\ell}$. It is natural to ask if the condition of low influences can be relaxed to the condition that the function has vanishing Fourier coefficients. Here we answer this question affirmatively. For the case of two functions $f$ and $g$, we further show that if $f,g$ have a noisy inner product that exceeds the gaussian bound, then the Fourier supports of their large coefficients intersect.