Pure point diffraction and cut and project schemes for measures: The smooth case

Abstract

We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of factor maps between the case of continuous and non-continuous weight functions.

30 pages

References in corpus (4)

• Pure Point Dynamical and Diffraction Spectra
• Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
• Deformation of Delone dynamical systems and pure point diffraction
• Dense Dirac combs in Euclidean space with pure point diffraction

Cited by in corpus (25)

• Dynamics on the graph of the torus parametrisation
• Pure Point spectrum for measure dynamical systems on locally compact Abelian groups
• Mathematical diffraction of aperiodic structures
• On pattern entropy of weak model sets
• Pure point diffraction and Poisson summation
• On the Fourier Analysis of Measures with Meyer Set Support
• Aperiodic order and pure point diffraction
• A short guide to pure point diffraction in cut-and-project sets
• On the Bragg Diffraction Spectra of a Meyer Set
• On Weighted Dirac Combs Supported Inside Model Sets
• Dynamical properties of almost repetitive Delone sets
• Periods and factors of weak model sets
• On norm almost periodic measures
• Leptin densities in amenable groups
• Modulated crystals and almost periodic measures
• On arithmetic progressions in model sets
• On sampling and interpolation by model sets
• Positive Positive Definite Discrete Strong Almost Periodic Measures and Bragg Diffraction
• 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
• Random fields on model sets with localized dependency and their diffraction