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



Is there an "analytical" version of Tao's uncertainty principle?
No. One just has to apply the standard example showing the classical uncertainty principle is sharp:


Let $f(a) = \sum\_{n \in \mathbb Z} e^{- \pi ( a+pn)^2 / p}$. Then $\hat{f}$ is proportional to $f$.


But $1-\epsilon$ of its mass is contained in an interval of width something like $O ( \sqrt{ p \log (1/\epsilon)})$, which is much smaller than $\epsilon p$.


