Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces

Abstract

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings.

11 pages, 2 figures; dedicated to Peter Kramer on the occasion of his 65th birthday

Cited by in corpus (34)

• What is a crystal?
• Dynamics on the graph of the torus parametrisation
• Diffraction of random tilings: some rigorous results
• Mathematical diffraction of aperiodic structures
• Spectral and topological properties of a family of generalised Thue-Morse sequences
• Kinematic Diffraction from a Mathematical Viewpoint
• On pattern entropy of weak model sets
• A note on palindromicity
• Linear Fractional p-Adic and Adelic Dynamical Systems
• Substitution Delone Sets with Pure Point Spectrum are Inter Model Sets
• Aperiodic crystals and beyond
• Computing modular coincidences
• A short guide to pure point diffraction in cut-and-project sets
• Diffractive point sets with entropy
• Diffraction of limit periodic point sets
• Duality of Model Sets Generated by Substitutions
• Scaling of diffraction intensities near the origin: Some rigorous results
• Surprises in aperiodic diffraction
• Close-packed dimers on the line: diffraction versus dynamical spectrum
• Weighted Dirac combs with pure point diffraction
• Diffraction Theory of Point Processes: Systems with Clumping and Repulsion
• On arithmetic progressions in non-periodic self-affine tilings
• Superquasicrystals: selfsimilar ordered structures with non-crystallographic point symmetries
• Hierarchical freezing in a lattice model
• On embedding of repetitive Meyer multiple sets into model multiple sets
• A Note on Transfinite M Theory and the Fine Structure Constant
• An ultrametric state space with a dense discrete overlap distribution: Paperfolding sequences
• Estimates of linearization discs in $p$-adic dynamics with application to ergodicity
• Diffraction of weighted lattice subsets
• Random fields on model sets with localized dependency and their diffraction
• A Quasiperiodic Tiling With 12-Fold Rotational Symmetry and Inflation Factor 1 + Sqrt(3)
• Selfdual Substitutions in Dimension One
• Self-dual tilings with respect to star-duality
• Lattice Substitution Systems and Model Sets