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Modulated crystals and almost periodic measures

Jeong-Yup Lee, Daniel Lenz, Christoph Richard, Bernd Sing, Nicolae Strungaru

Abstract

Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyse these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similarly to de Bruijn's analysis, we find stability with respect to almost periodic modulations.

Modulated crystals and almost periodic measures

Abstract

Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyse these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similarly to de Bruijn's analysis, we find stability with respect to almost periodic modulations.

Paper Structure

This paper contains 27 sections, 26 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sine modulated integers
  3. Overview
  4. The hull and its torus parametrization
  5. Diffraction of sine modulated integers
  6. Setting and notation
  7. Complex Radon measures
  8. Measure dynamical systems
  9. Mathematical diffraction theory
  10. Strongly almost periodic measures
  11. Cut-and-project schemes and model sets
  12. Lattice modulations
  13. Lattice modulations and deformations
  14. Modulations of modulated lattices
  15. Deformed weighted model sets
  16. ...and 12 more sections

Key Result

Lemma 2.1

The map $\mu: \mathbb{T}\to \Omega$ is a factor map, i.e., it is continuous onto and commutes with the natural $\mathbb{R}\space$-actions on $\mathbb{T}$ and on $\Omega$. If $\varepsilon<1/2$, then $\mu$ is a homeomorphism.

Figures (1)

  • Figure 1: Diffraction of sine modulated integers ($\varepsilon=1/20$, $\alpha=1/\tau^4$). Label $(m,n)$ corresponds to Bragg peak position $m+\alpha n$.

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • Remark 4.3
  • proof
  • Corollary 4.4
  • proof
  • ...and 53 more