Heisenberg uniqueness pairs for some algebraic curves in the plane

A Heisenberg uniqueness pair is a pair $(\Gamma, \Lambda)$, where $\Gamma$ is a curve and $\Lambda\subset\mathbb R^2$ such that any finite Borel measure $\mu$ which is supported on $\Gamma$ and absolutely continuous with respect to the arc length, whose Fourier transform $\widehat\mu$ vanishes on $\Lambda,$ implies $\mu=0.$ In this article, we shall investigate Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and the exponential curves. Further, we give a characterization of the Heisenberg uniqueness pairs corresponding to the four parallel lines. In the latter case, we observe the phenomenon of interlacing sets.