Heisenberg uniqueness pairs for the hyperbola

TL;DR

This paper investigates conditions under which a hyperbola and a perturbed lattice-cross form a Heisenberg uniqueness pair, extending previous results to rational perturbations and establishing a precise criterion based on the parameter .

Contribution

It characterizes Heisenberg uniqueness pairs for a hyperbola and a rationally perturbed lattice-cross, generalizing earlier work and providing exact bounds for .

Findings

Heisenberg uniqueness holds if and only if ;

Extension of previous results to rational perturbations of the lattice-cross

Explicit criterion for  based on the parameter p

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-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 $\beta\leq{p}.$