Pure point diffraction and Poisson summation

Abstract

We prove that the diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. This is achieved using Fourier analysis of unbounded measures on locally compact abelian groups as developed by Argabright and de Lamadrid. We also obtain diffraction results for certain classes of non-regular so-called weak model sets.

24 pages, submitted

References in corpus (2)

• Dynamics on the graph of the torus parametrisation
• Pure point diffraction and cut and project schemes for measures: The smooth case

Cited by in corpus (19)

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• Modulated crystals and almost periodic measures
• Leptin densities in amenable groups
• On arithmetic progressions in model sets
• On sampling and interpolation by model sets
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• Spectrum of weak model sets with Borel windows
• Pure point measures with sparse support and sparse Fourier--Bohr support
• On the Fourier Transformability of Strongly Almost Periodic Measures
• A note on measures vanishing at infinity
• On eigenmeasures under Fourier transform
• Why do Meyer sets diffract?
• Eberlein decomposition for PV inflation systems
• Diffraction of return time measures
• Measures, modular forms, and summation formulas of Poisson type