Searching for Regularity in Bounded Functions

Abstract: Given a function $f$ on $\mathbb{F}_2<sup>n$, 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 $f:\mathbb{F}_2ⁿ \to [-1,1]$, we show that there exists an affine subspace of dimension at least $\tilde\Omega(n<sup>{1/d!}k<sup>{-2})$, wherein all of $f$'s nontrivial Fourier coefficients become smaller than $2<sup>{-k}$. 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 $\Omega(dn<sup>{1/(d-1)})$. 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 $\mathbb{F}_2<sup>n$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.