Heisenberg uniqueness pairs in the plane. Three parallel lines

Abstract:

A Heisenberg uniqueness pair 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 bounded 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 characterize the Heisenberg uniqueness pairs for $\Gamma$ as being three parallel lines $\Gamma =\mathbb {R}\times \{\alpha ,\beta ,\gamma \}$ with $\alpha <\beta <\gamma ,$ $(\gamma -\alpha )/(\beta -\alpha )\in \mathbb {N}.$