Heisenberg uniqueness pairs for some algebraic curves in the plane

A Heisenberg uniqueness pair is a pair $\left(\Gamma, \Lambda\right)$, where $\Gamma$ is a curve and $\Lambda$ is a set in $\mathbb R^2$ such that whenever a finite Borel measure $\mu$ having support on $\Gamma$ which is absolutely continuous with respect to the arc length on $\Gamma$ satisfies $\hat\mu\vert_\Lambda=0,$ then it is identically $0.$ In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.