Weighted Dirac combs with pure point diffraction

By Michael Baake, Robert V. Moody. 2002

205 citations recorded by the source corpus.

A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.

Subfields: Geometry and Topology, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/93477fd0e2c87808a4a0806c992d4f7555c42b40

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In Geometry and Topology

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Citations in the corpus

On the Pisot Substitution Conjecture (2015, unknown): “See [19, 20, 76, 89, 101] for more on model sets.”
Pure Point Dynamical and Diffraction Spectra (2002, background): “(This approximation step requires some care; it is explained in detail in [1].”
Confirmation of Matheron's conjecture on the covariogram of a planar convex body (2007, background): “It turns out that quasicrystals can often be described by means of the so-called cut-and-project scheme; see [ BM04 ].”
Dynamics on the graph of the torus parametrization (2015, methodology): “As also the first part of Moody’s Theorem 3 holds, there is a full measure set X̂1 ⊆ X̂ such that configurations are generic for Q0M.”