Edit: we cannot find such an example. It would imply a negative solution to the KS${}_2$ conjecture which has now been proven by Marcus, Spielman, and Srivastava in this paper.

In fact, their solution implies that there always exists $S$ such that any $f$ supported on $S$ (or $S^c$) satisfies $\|\frac{1}{2}\hat{f} - \hat{f}|_T\|_2 \leq O(\sqrt{\epsilon})\cdot\|\hat{f}\|_2$.

Can we find, for every $\epsilon > 0$, an example of the following?

$\bullet$ a finite abelian group $G$

$\bullet$ a small subset $T$ of the dual group $\hat{G}$ (meaning $|T| \leq \epsilon |\hat{G}|$)

such that for any subset $S$ of $G$, there is a nonzero complex valued function $f$ supported either on $S$ or $S^c$ whose Fourier transform is approximately supported on $T$ (meaning $\|\hat{f} - \hat{f}|_T\|_2 \leq \epsilon \|\hat{f}\|_2$).

It seems unlikely, but I don't know any version of the uncertainty principle that would forbid this.

Two comments: (1) This arose out of discussions I had with Chuck Akemann about the Kadison-Singer problem. It wouldn't, as it stands, imply a negative solution to Kadison-Singer, but it would be close and (I believe) could probably be easily converted into a full negative solution. (2) Since either $S$ or $S^c$ has cardinality at least $|G|/2$, one could consider assuming $|S| \geq |G|/2$ and demanding that $f$ be supported on $S$. But I know that this makes the problem too hard. (Given $T$, I can always find an $S$ with $|S^c|/|G| = O(\sqrt{\epsilon})$ such that no nonzero function supported on $S$ has Fourier transform approximately supported on $T$.)