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

The setting is an ergodic dynamical system (X,μ)(X,μ) whose points are themselves uniformly discrete point sets ΛΛ in some space RdRd and whose group action is that of translation of these point sets by the vectors of RdRd. Steven Dworkin’s argument relates the diffraction of the typical point sets comprising XX to the dynamical spectrum of XX. In this paper we look more deeply at this relationship, particularly in the context of point processes.We show that there is an RdRd-equivariant, isometric embedding, depending on the scattering strengths (weights) that are assigned to the points of Λ∈XΛ∈X, that takes the L2L2-space of RdRd under the diffraction measure into L2(X,μ)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 μμ 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∈XΛ∈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 ϵϵ-dual characters of typical element of XX. The paper ends with a discussion of the Palm measure in the context of defining pattern frequencies.