2023/11/27 by Shanlin Huang, Jiaqi Yu, Huang, Shanlin +1
Engineering · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2311.15601
openalex publication_date 2023/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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'ıgez (Ann. of Math. 173(2),1507-1527, 2011, \citeHM) as a variant of the uncertainty principle for the Fourier transform. The main results of Hedenmalm and Motes-Rodr'ıgez concern the hyperbola Γε=\(x1, x2)∈ ℝ2, x1x2=ε\ (0≠ε∈ ℝ) and lattice-crosses Λαβ=(αℤ× \0\)∪(\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.