  Let $\mathbb{F}$ be a fixed finite field, and let $A \subset \mathbb{F}^n$.
It is a well-known fact that there is a subspace $V \leq \mathbb{F}^n$,
$\mbox{codim} V \ll_{\delta} 1$, and an $x$, such that $A$ is $\delta$-uniform
when restricted to $x + V$ (that is, all non-trivial Fourier coefficients of
$A$ restricted to $x + V$ have magnitude at most $\delta$). We show that if
$\mathbb{F} = \mathbb{F}_2$ then it is possible to take $x = 0$; that is, $A$
is $\delta$-uniform on a subspace $V \leq \mathbb{F}^n$. We give an example to
show that this is not necessarily possible when $\mathbb{F} = \mathbb{F}_3$.
  ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners
showed us a rather short argument he had found in 2013, giving a better bound
for our main theorem. We do not, therefore, intend to publish this note. The
example over $\mathbb{F}_3$ may still be of interest to some readers and so we
will not withdraw the paper from the arxiv.

Number Theory;Combinatorics