Perturbed lattice crosses and Heisenberg uniqueness pairs

This work focuses on two questions raised by H. Hedenmalm and A. Montes-Rodr\'iguez on Heisenberg Uniqueness Pairs for perturbed lattice crosses. The first of them deals with a complete characterization of $\beta>0$ for which, for a fixed $\theta \in \mathbb{R},$ the translated lattice cross $\Lambda_{\beta}^{\theta} = ((\mathbb{Z} + \{\theta\}) \times \{0\}) \cup (\{0\} \times \beta \mathbb{Z})$ satisfies that $(\Gamma,\Lambda_{\beta}^{\theta})$ is a Heisenberg Uniqueness Pair, where $\Gamma$ is the hyperbola in $\mathbb{R}^2$ with axes as asymptotes. We show that $(\Gamma,\Lambda_{\beta}^{\theta})$ is a Heisenberg Uniqueness Pair if and only if $\beta \le 1$, confirming a prediction made by Hedenmalm and Montes-Rodr\'iguez. Furthermore, under modified decay conditions on the measures under consideration, we are able to prove sharp results for when a perturbed lattice cross $\Lambda_{\bf A,B}$ is such that $(\Gamma,\Lambda_{\bf A,B})$ is a Heisenberg Uniqueness Pair. In particular, under such decay conditions, this solves another question posed by Hedenmalm and Montes-Rodr\'iguez. Our techniques run through the analysis of the action of the operator that maps the Fourier transform of an $L^1$ function $\psi$ to the Fourier transform of $t^{-2} \psi(1/t)$. In other words, we analyze the operator taking the restriction to the $x$-axis of a solution $u$ to the Klein-Gordon equation to its restriction to the $y$-axis. This operator turns out to be related to the action of the four-dimensional Fourier transform on radial functions, which enables us to use the framework and techniques of discrete uncertainty principles for the Fourier transform.