  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.
