Constant in Amrein-Berthier uncertainty principle

Question

Let $S,\Sigma$ in $\mathbb{R}^d$ be finite measure set. The Amrein-Berthier uncertainty principle states that there exists $C=C(S,\Sigma)>0$ such that for all $f\in L^2(\mathbb{R}^d)$, $\int_{\mathbb{R}^d} |f|^2\leq C \left(\int_{\mathbb{R}^d\setminus S} |f|^2+\int_{\mathbb{R}^d\setminus \Sigma} |\widehat{f}|^2\right)$.

Suppose $C$ is the best constant. It is known that the constant $C$ satisfies $C(S,\Sigma)\leq A e^{A |S||\Sigma|}$ with $A\geq 1$.
Do we also have that $C(S,\Sigma)\to 1$ when $|S||\Sigma|\to 0$ ? In this case, is there a simple equivalent or an estimate of $C(S,\Sigma)-1$ when $|S||\Sigma|\to 0$ ?

Answer 1

I remembered that this theorem is "essential" only when $|S||\Sigma|\geq1$. If $|S||\Sigma|<1$, one has an easier estimate (for $d=1$) $$||f||^2_{L^2}\leq\frac{1}{\sqrt{1-|S||\Sigma|}}\bigg(\int_{S^c}|f|^2+\int_{\Sigma^c}|\hat{f}|^2\bigg)$$ So you may see that $C\rightarrow1$ as $|S||\Sigma|$ goes to $0$. I think this should be resaonable also for higher dimension. This inequality is very easy(only few lines) and due to F.Nazarov, you may find the proof in his paper Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, at the end of section 2.2.

It is conjectured that in $\mathbb{R}^d$, the sharp estimate of $C$ is of the form $$C=Ce^{C(|S||\Sigma|)^{\frac{1}{d}}}$$ One can check the sharpness of $C$ by choosing $f$ Gaussian and $S$,$\Sigma$ balls centered at the origin. This sharp estimate has been proved when at least one of $S$ and $\Sigma$ is convex, but still open for general $S$ and $\Sigma$. You may see this in P.Jaming's paper Nazarov's uncertainty principles in higher dimension. This is one of my topic in research but sorry I do not know any other esimates on $C$.