Searching for Regularity in Bounded Functions

TL;DR

This paper investigates the structure of bounded functions over finite fields, identifying large affine subspaces where Fourier coefficients are small, with implications for pseudorandomness and affine dispersers.

Contribution

It establishes bounds on the size of affine subspaces where Fourier coefficients diminish for degree d functions, and provides explicit examples and characterizations related to these properties.

Findings

Existence of large affine subspaces with small Fourier coefficients for degree d functions.

Construction of degree d functions with large Fourier coefficients on any large affine subspace.

Connections to parity kill number and affine dispersers.

Abstract

Given a function $f$ on $\mathbb{F}_2^n$, we study the following problem. What is the largest affine subspace $\mathcal{U}$ such that when restricted to $\mathcal{U}$, all the non-trivial Fourier coefficients of $f$ are very small? For the natural class of bounded Fourier degree $d$ functions $f:\mathbb{F}_2^n \to [-1,1]$, we show that there exists an affine subspace of dimension at least $ \tilde\Omega(n^{1/d!}k^{-2})$, wherein all of $f$'s nontrivial Fourier coefficients become smaller than $ 2^{-k}$. To complement this result, we show the existence of degree $d$ functions with coefficients larger than $2^{-d\log n}$ when restricted to any affine subspace of dimension larger than $\Omega(dn^{1/(d-1)})$. In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of $\mathbb{F}_2^n$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.