Heisenberg uniqueness pairs for some algebraic curves in the plane

A Heisenberg uniqueness pair is a pair (Γ,Λ), where Γ is a curve and Λ is a set in R2 such that whenever a finite Borel measure μ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfies μˆ|Λ=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.