Heisenberg uniqueness pairs for the hyperbola

Let $\mathrm{\Gamma }$ be the hyperbola $\{(x,y)\in {\mathbb{R}}^2:xy=1\}$ and ${\mathrm{\Lambda }}_{\beta }$ 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 $(\mathrm{\Gamma },{\mathrm{\Lambda }}_{\beta })$ is a Heisenberg uniqueness pair if and only if $\beta \le 1.$ In this paper, we show that for a rational perturbation of ${\mathrm{\Lambda }}_{\beta },$ namely $${\mathrm{\Lambda }}_{\beta }^{\theta }=((\mathbb{Z}+\{\theta \})\times \{0\})\cup (\{0\}\times \beta \mathbb{Z}),$$ where $\theta =1\mathrm{/}p,\text{ for some }p\in \mathbb{N}$ and $\beta$ is a positive real, the pair $(\mathrm{\Gamma },{\mathrm{\Lambda }}_{\beta }^{\theta })$ is a Heisenberg uniqueness pair if and only if $\beta \le p\mathrm{.}$