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Heisenberg uniqueness pairs for the hyperbola
Author(s) -
Giri Deb Kumar,
Rawat Rama
Publication year - 2021
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12391
Subject(s) - hyperbola , mathematics , uniqueness , mathematical analysis , geometry
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 .