Heisenberg uniqueness pairs for some algebraic curves and surfaces

Let $X(\mathrm{\Gamma })$ be the space of all finite Borel measure $\mu$ in $\mathbb R^2$whichissupportedonthecurve$\Gamma$ and absolutely continuous with respect to the arc length of $\mathrm{\Gamma }$. For $\mathrm{\Lambda }\subset {\mathbb{R}}^2,$ the pair $(\mathrm{\Gamma },\mathrm{\Lambda })$ is called a Heisenberg uniqueness pair for $X(\mathrm{\Gamma })$ if any $\mu \in X(\mathrm{\Gamma })$ satisfies $\widehat{\mu }{\mathrm{\mid }}_{\mathrm{\Lambda }}=0,$ implies $\mu =0.$ We explore the Heisenberg uniqueness pairs corresponding to the cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines. We observe that the size of the determining sets $\mathrm{\Lambda }$ for $X(\mathrm{\Gamma })$ depends on the number of lines and their irregular distribution that further relates to a phenomenon of interlacing of certain trigonometric polynomials.