Heisenberg uniqueness pairs for the hyperbola

Abstract

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-Rodríguez says that $\left(\Gamma ,{\Lambda }_{\beta }\right)$ is a Heisenberg uniqueness pair if and only if $\beta \le 1.$ In this paper, we show that for a rational perturbation of ${\Lambda }_{\beta },$ namely \[Λ_β^θ=\left((\mathbb Z+\{θ\})\times\{0\}\right)\cup\left(\{0\}\timesβ\mathbb Z\right),\] 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 $\beta \le p.$

12 pages