Fourier uniqueness sets and the Klein-Gordon equation

TL;DR

This paper investigates Fourier uniqueness sets related to the Klein-Gordon equation, identifying critical densities and revealing non-uniqueness phenomena linked to the Nielsen spiral.

Contribution

It extends the theory of Heisenberg uniqueness pairs to the Klein-Gordon context and uncovers new non-uniqueness results at critical densities.

Findings

Identified the critical density for uniqueness on a hyperbola branch.

Discovered non-uniqueness for a one-dimensional solution space at critical density.

Linked non-uniqueness solutions to the Nielsen spiral phenomenon.

Abstract

We generalize the study of Heisenberg uniqueness pairs considered in earlier work with Alfonso Montes-Rodriguez. We also find the critical density in the case of one branch of the hyperbola. In the critical case there is non-uniqueness for a one-dimensional space of solutions. These solutions cannot vanish at any other point of the axes of the characteristic lattice-cross, due to an unexpected connection with the Nielsen spiral.