Heisenberg uniqueness pairs and the Klein-Gordon equation

Abstract

A Heisenberg uniqueness pair (HUP) is a pair $(\Gamma,\Lambda)$, where $\Gamma$ is a curve in the plane and $\Lambda$ is a set in the plane, with the following property: any finite Borel measure $\mu$ in the plane supported on $\Gamma$, which is absolutely continuous with respect to arc length, and whose Fourier transform $\widehat\mu$ vanishes on $\Lambda$, must automatically be the zero measure. We prove that when $\Gamma$ is the hyperbola $x_1x_2=1$ %, and $\Lambda$ is the lattice-cross \[\Lambda=(\alpha\mathbb{Z}\times\{0\})\cup(\{0\}\times\beta\mathbb{Z}),\] where $\alpha,\beta$ are positive reals, then $(\Gamma,\Lambda)$ is an HUP if and only if $\alpha\beta\le1$; in this situation, the Fourier transform $\widehat\mu$ of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that \[{\mathrm e}^{\pi{\mathrm i} \alpha n t},\,\,{\mathrm e}^{\pi{\mathrm i}\beta n/t},\qquad n\in\mathbb{Z},\] span a weak-star dense subspace in $L^\infty(\mathbb{R})$ if and only if $\alpha\beta\le1$. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.