Heisenberg uniqueness pairs for the hyperbola

Let Γ be the hyperbola { ( x , y ) ∈ R 2 : x y = 1 } and Λ β be the lattice‐cross defined byΛ β = ( Z × { 0 } ) ∪ ( { 0 } × β Z )in R 2 , where β is a positive real. A result of Hedenmalm and Montes‐Rodríguez says that ( Γ , Λ β ) is a Heisenberg uniqueness pair if and only if β ⩽ 1 . In this paper, we show that for a rational perturbation of Λ β , namelyΛ β θ = ( Z + { θ } ) × { 0 } ∪ { 0 } × β Z , where θ = 1 / p , for some p ∈ N and β is a positive real, the pair ( Γ , Λ β θ ) is a Heisenberg uniqueness pair if and only if β ⩽ p .