Let $\mathcal F$ denotes the Fourier transform $\mathcal{F} :L^2(\mathbb R)\rightarrow L^2(\mathbb R)$ and $E, \Sigma$ be two measurable sets in $\mathbb R$.

The pair $(E,\Sigma)$ is called a weakly annihilating pair, if for any $f \in L^2(\mathbb{R})$, $support (f) \subseteq E$, $support(\mathcal F f)\subseteq \Sigma$, implies $f \equiv 0$.

The pair $(E,\Sigma)$ is called a strongly annihilating pair, if there exists a constant $C$ such that for any $f \in L^2(\mathbb{R})$,

$$\|f\|_2^2 \leq C \left(\int_{\mathbb R \setminus E} |f|^2 dx + \int_{\mathbb R \setminus \Sigma} |\mathcal F f|^2 d\xi \right).$$ The notion of annihilating pair arises in the study of uncertainty property in Fourier Analysis. For example Benedicks's Theorem says if $E$, $\Sigma$ are both sets of finite measures then they form a weakly annihilating pair, whereas Theorem of Amrein and Berthier says they form a strongly annihilating pair.

I am looking for examples of

~~ 1) A weakly annihilating pair which is $\underline{not}$ a strongly annihilating pair. ~~ ( Willie Wong has already answered this and I realise this was rather easy and I should have been able to figure it out myself, so my apologies.)

1') Sets $E$, $\Sigma$ both have infinite measure such that $(E,\Sigma)$ is a strongly annihilating pair.

2) Sets $E$, $\Sigma$ such that $E^c$ and $\Sigma^c$ have nonzero measure and $(E,\Sigma)$ is $\underline{not}$ a weakly annihilating pair.

I realise the standard reference for this topic is the book by Havin and Jöricke, which unfortunately our library does not have a copy of!! Is there any alternative reference someone can suggest ?

Thankyou.