Uniform Distribution in Model Sets

2002/03/01 by Robert V. Moody · 6 citations
Mathematics · Computer Science · Materials Science · #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #Quasicrystal Structures and Properties

paper · pdf · doi:10.4153/cmb-2002-015-3

Abstract

Abstract We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining themodel set ismeasurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

Cited by

• Model sets with positive entropy in Euclidean cut and project schemes
• Weighted model sets and their higher point-correlations
• Model sets with precompact Borel windows
• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
• Exact renormalisation for patch frequencies in inflation systems
• Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks

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