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

mostof the mass of $f$ and $\hat f$ (rather thanallof 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