A dynamical system approach to Heisenberg uniqueness pairs

Abstract

Let Λ be a set of lines in R² that intersect at the origin. For a smooth curve Γ ⊂ R², we denote by AC(Γ) the subset of finite measures on Γ that are absolutely continuous with respect to arc length on Γ. For μ ∈ AC(Γ), $\hat \mu$ denotes the Fourier transform of μ. Following Hedenmalm and Montes-Rodríguez, we say that (Γ,Λ) is a Heisenberg uniqueness pair if μ ∈ AC(Γ) is such that $\hat \mu$ = 0 on Λ implies μ = 0. The aim of this paper is to provide new tools to establish this property. To do so, we reformulate the fact that $\hat \mu$ vanishes on Λ in terms of an invariance property of μ induced by Λ. This leads us to a dynamical system on Γ generated by Λ. In many cases, the investigation of this dynamical system allows us to establish that (Γ,Λ) is a Heisenberg uniqueness pair. This way we both unify proofs of known cases (circle, parabola, hyperbola) and obtain many new examples. This method also gives a better geometric intuition as to why (Γ,Λ) is a Heisenberg uniqueness pair. As a side result, we also give the first instance of a positive result in the classical Cramér-Wold theorem where finitely many projections suffice to characterize a measure (under strong support constraints).