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Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

In this article, we prove that (unit sphere, non-harmonic cone) is a Heisenberg uniqueness pair for the symplectic Fourier transform on $\mathbb C^n.$ We derive that spheres as well as non-harmonic cones are determining sets for the spectral projections of the finite measure supported on the unit sphere. Further, we prove that if the Fourier transform of a finitely supported function on step two nilpotent Lie group is of arbitrary finite rank, then the function must be zero. The latter result correlates to the annihilating pair for the Weyl transform.