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Weighted Dirac combs with pure point diffraction. (English) Zbl 1188.43008
Summary: 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 theory of 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.

MSC:
43A80 Analysis on other specific Lie groups
52C23 Quasicrystals and aperiodic tilings in discrete geometry
82D25 Statistical mechanics of crystals
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