TITLE: Is there an "analytical" version of Tao's uncertainty principle?
QUESTION [17 upvotes]: Let $p$ be a prime. For $f: \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{C}$ let its Fourier transform be:
$$\hat f(n) = \frac{1}{\sqrt{p}}\sum_{x \in \mathbb{Z}/p \mathbb{Z}} f(x)\, e\left(\frac{-xn}{p}\right)$$
Terence Tao proves in "An uncertainty principle for cyclic groups of prime order" that for $f: \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{C}$ one has:
$$|S_1| + |S_2|\ge p+1$$
If $f$ is supported in $S_1$ and $\hat f$ is supported in $S_2$. This is much stronger than the traditional bound:
$$|S_1||S_2| \ge p$$

Can we prove an approximate version of this inequality, say, in which we lower bound
$$|S_1| + |S_2|$$
assuming only that $S_1$ and $S_2$ contain most of the mass of $f$ and $\hat f$ (rather than all of the mass)?

The proof of the lower bound to $|S_1||S_2|$ is "only analytical", so it caries over under this weaker assumption, but Tao's lower bound for $|S_1|+|S_2|$ relies strongly on algebra, in that it is important that $f$ and $\hat f$ are exactly $0$ outside the support, so the proof doesn't immediately carry over.
Is there such an approximate version of Tao's uncertainty principle?
For a concrete case, if the Fourier transform is normalized so that $\sum |f(x)|^2 = \sum |\hat f(x)|^2 = 1$, can we rule out the possibility that $\sum_{x \in S_1} |f(x)|^2 > 1-\epsilon$, and  $\sum_{x\in S_2} |\hat f(x)|^2 > 1-\epsilon$ with $|S_1|, |S_2| < \epsilon p$?
For the applications that I am interested I need something quite weaker, but I couldn't really find a counter-example to the above possibility.
Thanks in advance

REPLY [17 votes]: One does not need Gaussians in the finite case, just take $f$ to be the indicator function of the interval $[-(n-1),n-1]\subset\mathbb F_p$. A simple computation gives
  $$ |\hat f(x)| = \frac1{\sqrt p} \frac{|\sin\pi(2n-1)x/p|}{|\sin\pi(x/p)|} < \frac1{2\sqrt p\|x/p\|}, $$
where $\|x/p\|$ is the distance to $x/p$ from the nearest integer. As a result, 
  $$ \sum_{x=m}^{p-m} |\hat f(x)|^2 \le \frac p{2m} = \frac p{2mn} \sum_{x=0}^{p-1} |\hat f(x)|^2. $$
Taking, say, $m,n\approx\sqrt{\varepsilon^{-1}p}$, you have both $f$ and $\hat f$ concentrated on intervals of length about $2\sqrt{\varepsilon^{-1}p}$.

A historical point worth clarification. The inequality $|{\rm supp}\, f|+|{\rm supp}\,\hat f|\ge p+1$ was in fact proved first by Andras Biro, who has contributed it as a problem to the 1998 Schweitzer competition (Problem 3). Bearing in mind that Tao's paper appeared in 2005, I think it would be most reasonable to call it the Biro-Tao inequality.