Heisenberg Uniqueness Pairs and the wave equation

Given a curve $\Gamma$ and a set $\Lambda$ in the plane, the concept of the Heisenberg uniqueness pair $(\Gamma, \Lambda)$ was first introduced by Hedenmalm and Motes-Rodr\'{\i}gez (Ann. of Math. 173(2),1507-1527, 2011, \cite{HM}) as a variant of the uncertainty principle for the Fourier transform. The main results of Hedenmalm and Motes-Rodr\'{\i}gez concern the hyperbola $\Gamma_{\epsilon}=\{(x_1, x_2)\in \mathbb{R}^2,\, x_1x_2=\epsilon\}$ ($0\ne\epsilon\in \mathbb{R}$) and lattice-crosses $\Lambda_{\alpha\beta}=(\alpha\mathbb{Z}\times \{0\})\cup(\{0\}\times \beta\mathbb{Z})$ ($\alpha, \beta>0$), where it's proved that $(\Gamma_{\epsilon}, \Lambda_{\alpha\beta})$ is a Heisenberg uniqueness pair if and only if $\alpha\beta\leq 1/|\epsilon|$. In this paper, we aim to study the endpoint case (i.e., $\epsilon=0$ in $\Gamma_{\epsilon}$) and investigate the following problem: what's the minimal amount of information required on $\Lambda$ (the zero set) to form a Heisenberg uniqueness pair? When $\Lambda$ is contained in the union of two curves in the plane, we give characterizations in terms of some dynamical system conditions. The situation is quite different in higher dimensions and we obtain characterizations in the case that $\Lambda$ is the union of two hyperplanes.