The notion of Heisenberg uniqueness pair has been introduced by Hedenmalm and Montes-Rodriguez (Ann. of Math. 2011) as a version of the uncertainty principle, that is, a nonzero function and its Fourier transform both cannot be too small simultaneously. Let $\Gamma$ be a smooth curve or finite disjoint union of smooth curves in the plane and $\Lambda$ be any subset of the plane. Let $\mathcal X(\Gamma)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $\Gamma$ and are absolutely continuous with respect to the arc length measure on $\Gamma.$ Let $\mathcal{AC}(\Gamma,\Lambda)=\{\mu\in \mathcal{X}(\Gamma) : \widehat\mu|_{\Lambda}=0\},$ then we say that $\Lambda$ is a Fourier uniqueness set for $\Gamma$ or $(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair, if $\mathcal{AC}(\Gamma,\Lambda)={0}.$

In this talk, we will discuss the following: Let $\Gamma$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $\Lambda_\beta^\theta$ be the lattice-cross defined by \begin{equation} \Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right) \cup \left(\{0\}\times\beta\mathbb Z\right), \end{equation} where $\beta$ is a positive real and $\theta=1/{p}$, for some $p\in\mathbb N,$ then $\left(\Gamma,\Lambda_\beta^\theta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq{p}.$ Moreover, the space $\mathcal{AC}\left(\Gamma,\Lambda_\beta^\theta\right)$ is infinite-dimensional provided $\beta>p.$