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

Abstract

Abstract. 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 theWe 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 2-point, 3-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 ǫ-dual characters of typical element of X. The paper ends with a discussion of the Palm measure in the context of defining pattern frequencies.

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

“…First steps are contained in [33,4,28,40,41,10,21,11] and indicate that both qualitative and quantitative results are possible, though some further development of the theory is needed.…”

Recent developments in the mathematics of diffraction

“…Thm. 4.2 as stated here appears in [9]. Theorem 4.3 [13] With ðX; mÞ as above, m is a pure point measure (and hence L 2 X is almost surely pure point diffractive) if and only if every measurable subset B of X satisfies the condition that for all E > 0 the set…”

“…; x n 2 L \ C R . This ostensibly belongs to L, but the theory shows that this limit exists and is the same for almost all L in X, so we can ignore its apparent dependence on a particular element of X, see [5,13] for n ¼ 2, [9] in general.…”

“…Recent developments in the mathematics of diffraction 799 9 Although H is not uniquely determined by L, it is possible to arrange things so that W generates H as a group and there are no non-zero t 2 H with t þ W ¼ W. With these conditions H is unique up to isomorphism (e.g. for the Penrose tilings it is R 2 Â Z=5Z).…”

A note on measures vanishing at infinity

“…Pure point diffraction was characterised in terms of almost periodicity of the autocorrelation measure [15,19,24,34,46], in terms of almost periodicity of the underlying structure [33] or in terms of the pure point dynamical spectrum [12,25]. Various systems with pure point diffraction, such as regular model sets and weighted Dirac combs with Meyer set support [15,21,28,37,38,39,49,41], weak model sets of maximal density [11,23], stationary processes [17,27] and various deformations [13,29], or almost periodic measures [30] have been studied. For more general overviews of these results we recommend [5,4,26,14,7].…”