Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

The setting is an ergodic dynamical system (X,μ) whose points are themselves uniformly discrete point sets Λ in some space R^d and whose group action is that of translation of these point sets by the vectors of R^d. Steven Dworkin's argument relates the diffraction of the typical point sets comprising X to the dynamical spectrum of X. In this paper we look more deeply at this relationship, particularly in the context of point processes. We show that there is an R^d-equivariant, isometric embedding, depending on the scattering strengths (weights) that are assigned to the points of Λ∈X, that takes the L²-space of R^d under the diffraction measure into L²(X,μ). We examine the image of this embedding and give a number of examples that show how it fails to be surjective. We show that full information on the measure μ is available from the weights and set of all the correlations (that is, the two-point, three-point, …, correlations) of the typical point set Λ∈X. We develop a formalism in the setting of random point measures that includes multi-colour point sets, and arbitrary real-valued weightings for the scattering from the different colour types of points, in the context of Palm measures and weighted versions of them. As an application we give a simple proof of a square-mean version of the Bombieri-Taylor conjecture, and from that we obtain an inequality that gives a quantitative relationship between the autocorrelation, the diffraction, and the ɛ-dual characters of typical element of X. The paper ends with a discussion of the Palm measure in the context of defining pattern frequencies.