Since the conception of quantum mechanics, Uncertainty Principles have played a central role in Analysis and Partial Differential Equations. Indeed, from Heisenberg, through Hardy, and all the way to unique continuation, this rich topic has expanded to several new, exciting direction throughout the past century. 

One of the most notable of such directions is the emerging area of Fourier Uniqueness Pairs. In spite of the celebrated contribution of M. Viazovska to the field, inspiring many questions with her Fourier analysis solution to the sphere packing problem in dimensions 8 and 24, a preceding work by H. Hedenmalm and A. Montez-Rodríguez had incidentally already touched upon the same sort of results years prior, by establishing a link between Fourier analysis, the Klein-Gordon equation and dynamical systems.

In this talk, we will explore this seminal controbution by Hedenmalm and Montes-Rodríguez, its relationship to Fourier interpolation, and some of the conjectures posed by those authors.