Aperiodic Order and Spherical Diffraction

We introduce model sets in arbitrary locally compact second countable (lcsc) groups, generalizing Meyer’s definition of a model set in a locally compact abelian group. We then provide a new formulation of diffraction theory, which unlike the classical formulation does not involve Følner sets and thus generalizes to point sets in non-amenable lcsc groups. We focus on the case of lcsc groups admitting a Gelfand pair and on the spherical part of the diffraction. Using this approach we obtain explicit formulas for the auto-correlation and diffraction of model sets. Our diffraction formula generalizes the spherical trace formula in a similar way as the abelian diffraction formula generalizes the Poisson summation formula. We deduce that a model set has pure point spherical diffraction provided the underlying lattice is cocompact.