Substitution Delone Sets with Pure Point Spectrum are Inter Model Sets

Abstract

The paper establishes an equivalence between pure point diffraction and certain types of model sets, called inter model sets, in the context of substitution point sets and substitution tilings. The key ingredients are a new type of coincidence condition in substitution point sets, which we call algebraic coincidence, and the use of a recent characterization of model sets through dynamical systems associated with the point sets or tilings.

29pages; revised version with updates

References in corpus (7)

• Pure Point Dynamical and Diffraction Spectra
• Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
• Characterizations of model sets by dynamical systems
• Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
• Computing modular coincidences
• A Characterization of Model Multi-colour Sets
• Pseudo-self-affine tilings in R^d

Cited by in corpus (8)

• Spectral and topological properties of a family of generalised Thue-Morse sequences
• Aperiodic order and pure point diffraction
• Overlap coincidence to strong coincidence in substitution tiling dynamics
• On arithmetic progressions in non-periodic self-affine tilings
• Algorithm for determining pure pointedness of self-affine tilings
• The computation of overlap coincidence in Taylor-Socolar substitution tiling
• Inter model sets in ${\mathbb{R}}^d$ are model sets
• Conjugacies of model sets