Heisenberg uniqueness pairs for the hyperbola

Let $\Gamma$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $\Lambda_\beta$ be the lattice-cross defined by $\Lambda_\beta=\left(\mathbb Z\times\{0\}\right)\cup\left(\{0\}\times\beta\mathbb Z\right)$ in $\mathbb R^2,$ where $\beta$ is a positive real. A result of Hedenmalm and Montes-Rodr\'iguez says that $\left(\Gamma,\Lambda_\beta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq1.$ In this paper, we show that for a rational perturbation of $\Lambda_\beta,$ namely $$\Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right)\cup\left(\{0\}\times\beta\mathbb Z\right),$$ where $\theta=1/{p},~\text{for some}~{p}\in\mathbb N$ and $\beta$ is a positive real, the pair $\left(\Gamma,\Lambda_\beta^\theta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq{p}.$