Almost Periodic Measures and Long-Range Order in Meyer Sets

Abstract

The main result of this paper is that the diffraction pattern of any Meyer set with a well-defined autocorrelation has a relatively dense set of Bragg peaks. In the second part of the paper we provide a necessary and sufficient condition for a positive pure point measure to have a continuous Fourier transform. In particular, one can get a necessary and sufficient condition for a point set to have no Bragg peaks in its diffraction.

Reminiscences About a Chemistry Nobel Prize Won with Metallurgy: Comments on D. Shechtman and I. A. Blech; Metall. Trans. A, 1985, vol. 16A, pp. 1005–12

“…Amusingly enough, this property was shown in 2005 to be one of the main ingredients in the Strungaru theorem that states that a Delaunay distribution of atoms has a sharp diffractive component in its Fourier spectrum if the pair interatomic vectors form a uniformly discrete set. [9] The Blech model has been the starting point of an enormous body of work on random tilings, especially in the United States, mainly by theoretical physicists.…”

Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number

“…On the other hand, the function giving the intensity per diffracting site of Z β is defined as [10] it can be shown that the values of (4.21) at the points of the pure-point spectrum of µ are the intensities of the Bragg peaks; see for instance Definition 3.1 in Strungaru [39]. This statement originates in the so-called…”

“…Under such assumptions, the autocorrelation is even the sum of two pure point measures (Proposition 3.3 in [39] show a need for introducing new theories to describe the structure of this spectrum. By Section 3 the elements in Z β are in one-to-one correspondence with elements of (−β, β) (Proposition 3.1) by the * -operation.…”

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

“…A particularly relevant class of point sets in the theory of aperiodic order are Meyer sets, which are relatively dense sets Λ such that Λ − Λ is uniformly discrete. Such sets always have a diffraction measure with a non-trivial pure point part, with a relatively dense supporting set [59,2], despite the fact that Meyer sets can have entropy 2 . If modified by a family of random measures according to Theorem 2, the resulting diffraction still shows the original diffraction with its non-trivial pure point component, modulated by the function E Q (Ω)…”