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A Heisenberg uniqueness pair is a pair ( , ) , where is a curve in the plane and is a set in the plane, with the following property: any bounded Borel measure µ in the plane supported on , which is absolutely continuous with respect to arc length, and whose Fourier transform b µ vanishes on , must automatically be the zero measure. We characterize the Heisenberg uniqueness pairs for being three parallel lines = R◊{,, } with < < , ( )/( )2 N.