  Let G be a finite abelian group. We examine the discrepancy between subspaces
of l^2(G) which are diagonalized in the standard basis and subspaces which are
diagonalized in the dual Fourier basis. The general principle is that a Fourier
subspace whose dimension is small compared to |G| = dim(l^2(G)) tends to be far
away from standard subspaces. In particular, the recent positive solution of
the Kadison-Singer problem shows that from within any Fourier subspace whose
dimension is small compared to |G| there is standard subspace which is
essentially indistinguishable from its orthogonal complement.
