Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations

Abstract

The paper studies the relationship between diffraction and dynamics for uniformly discrete ergodic point processes in real spaces. This relationship takes the form of an isometric embedding of two L^2 spaces. Diffraction (or equivalently the 2-point correlations) usually cannot determine the dynamics entirely, but we prove that knowledge of all the higher correlations (2-point, 3-point, ...) does. A square-mean form of the Bombieri-Taylor conjecture is proved. A quantitative relation between autocorrelation, diffraction, and epsilon dual characters is derived. Most results of the paper are proved in the setting of multi-colour points and assignable scattering strengths.