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.

翻译：给定 $\mathbb{F}_2^n$ 上的函数 $f$，我们研究如下问题：最大的仿射子空间 $\mathcal{U}$ 的维数是多少，使得当 $f$ 限制在 $\mathcal{U}$ 上时，所有非平凡傅里叶系数都非常小？对于自然的有界傅里叶度数 $d$ 的函数类 $f:\mathbb{F}_2^n \to [-1,1]$，我们证明存在一个维数至少为 $\tilde\Omega(n^{1/d!}k^{-2})$ 的仿射子空间，在该子空间上 $f$ 的所有非平凡傅里叶系数都小于 $2^{-k}$。为补充此结果，我们证明存在度数为 $d$ 的函数，当其限制在维数大于 $\Omega(dn^{1/(d-1)})$ 的任何仿射子空间上时，其系数仍大于 $2^{-d\log n}$。此外，我们给出了具有类似但较弱性质函数的显式例子。在证明过程中，我们提供了函数限制在 $\mathbb{F}_2^n$ 子空间上的傅里叶系数的多种刻画，这些刻画可能在其他上下文中有所助益。最后，我们强调这些结果在奇偶性消除数与仿射分散器中的应用与联系。