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 the model set is measurable 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.

MSC Classifications:

52C23, 11K70, 28D05, 37A30 show english descriptions Quasicrystals, aperiodic tilings
Harmonic analysis and almost periodicity
Measure-preserving transformations
Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 52C23 - Quasicrystals, aperiodic tilings
11K70 - Harmonic analysis and almost periodicity
28D05 - Measure-preserving transformations
37A30 - Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}

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