Abstract. A Heisenberg uniqueness pair is a pair (Γ, Λ), where Γ is a curve and Λ is a set in R 2 such that whenever a finite Borel measure µ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfiesμ| 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.

Let normalΓ be the hyperbola {false(x,yfalse)∈double-struckR2:xy=1} and Λβ be the lattice‐cross defined by normalΛβ=false(double-struckZ×{0}false)∪false({0}×βdouble-struckZfalse) in R2, where β is a positive real. A result of Hedenmalm and Montes‐Rodríguez says that (Γ,normalΛβ) is a Heisenberg uniqueness pair if and only if β⩽1. In this paper, we show that for a rational perturbation of Λβ, namely normalΛβθ=false(double-struckZ+false{θfalse}false)×false{0false}∪false{0false}×βdouble-struckZ,where θ=1/p,forsomep∈N and β is a positive real, the pair (Γ,normalΛβθ) is a Heisenberg uniqueness pair if and only if β⩽p.

In this article, we prove that if the group Fourier transform of certain integrable functions on the Heisenberg motion group (or step two nilpotent Lie groups) is of finite rank, then the function is identically zero. These results can be thought as an analogue to the Benedicks theorem that dealt with the uniqueness of the Fourier transform of integrable functions on the Euclidean spaces.

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