Perturbed lattice crosses and Heisenberg uniqueness pairs

Abstract

This work focuses on two questions raised by H. Hedenmalm and A. Montes-Rodríguez on Heisenberg Uniqueness Pairs for perturbed lattice crosses. The first of them deals with a complete characterization of $β>0$ for which, for a fixed $θ\in \mathbb{R},$ the translated lattice cross $Λ_β^θ = ((\mathbb{Z} + \{θ\}) \times \{0\}) \cup (\{0\} \times β\mathbb{Z})$ satisfies that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair, where $Γ$ is the hyperbola in $\mathbb{R}^2$ with axes as asymptotes. We show that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair if and only if $β\le 1$, confirming a prediction made by Hedenmalm and Montes-Rodríguez. Furthermore, under modified decay conditions on the measures under consideration, we are able to prove sharp results for when a perturbed lattice cross $Λ_{\bf A,B}$ is such that $(Γ,Λ_{\bf A,B})$ is a Heisenberg Uniqueness Pair. In particular, under such decay conditions, this solves another question posed by Hedenmalm and Montes-Rodríguez. Our techniques run through the analysis of the action of the operator that maps the Fourier transform of an $L^1$ function $ψ$ to the Fourier transform of $t^{-2} ψ(1/t)$. 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.