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 $|S||\Sigma|\to 0$ ?



Constant in Amrein-Berthier uncertainty principle
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 $C$.


