Given a curve Γ and a set Λ in the plane, the concept of the
Heisenberg uniqueness pair (Γ,Λ) was first introduced by
Hedenmalm and Motes-Rodr\'{\i}gez (Ann. of Math. 173(2),1507-1527, 2011,
\cite{HM}) as a variant of the uncertainty principle for the Fourier transform.
The main results of Hedenmalm and Motes-Rodr\'{\i}gez concern the hyperbola
Γϵ​={(x1​,x2​)∈R2,x1​x2​=ϵ}
(0î€ =ϵ∈R) and lattice-crosses
Λαβ​=(αZ×{0})∪({0}×βZ) (α,β>0), where it's proved that
(Γϵ​,Λαβ​) is a Heisenberg uniqueness pair if
and only if αβ≤1/∣ϵ∣.
In this paper, we aim to study the endpoint case (i.e., ϵ=0 in
Γϵ​) and investigate the following problem: what's the minimal
amount of information required on Λ (the zero set) to form a Heisenberg
uniqueness pair? When Λ is contained in the union of two curves in the
plane, we give characterizations in terms of some dynamical system conditions.
The situation is quite different in higher dimensions and we obtain
characterizations in the case that Λ is the union of two hyperplanes.Comment: 24 page