Heisenberg uniqueness pairs and the wave equation

Abstract The concept of the Heisenberg uniqueness pair was introduced by Hedenmalm and Montes‐Rodríguez as a variant of the uncertainty principle for the Fourier transform. The main results in Hedenmalm and Montes‐Rodríguez (Ann. of Math. (2) 173 (2011), 1507–1527) concern the hyperbola () and lattice‐crosses (), where it is proved that is a Heisenberg uniqueness pair if and only if . In this paper, we study the endpoint case ( in ) and investigate the minimal information required on the zero set to form such a pair. When is contained in the union of two curves in the plane, we provide characterizations using dynamical system conditions. The situation differs in higher dimensions, where we obtain characterizations when is the union of two hyperplanes.