Speaker
Description
Michael Whitmeyer and Siddharth Iyer
Abstract: Given a bounded function $f$ on $\mathbb F_2^n$, this work seeks a large affine subspace $\mathcal U$ such that $f$, when restricted to $\cal U$, has small nontrivial Fourier coefficients.
It is not hard to see that a random function is such that all its Fourier coefficients are extremely small already, so we focus on a natural class structured functions.
For the natural class of 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.