Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectrum of weak model sets with Borel windows

Gerhard Keller, Christoph Richard, Nicolae Strungaru

Abstract

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article "Periods and factors of weak model sets" we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody's work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

Spectrum of weak model sets with Borel windows

Abstract

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article "Periods and factors of weak model sets" we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody's work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

Paper Structure

This paper contains 11 sections, 15 theorems, 30 equations.

Table of Contents

  1. The result
  2. Proof ingredients
  3. Moody's uniform distribution theorem
  4. Haar periods and periods
  5. Proofs
  6. Haar aperiodic windows
  7. General windows
  8. Consequences for diffraction
  9. A class of examples
  10. Results for the general setting
  11. An example from ${\mathscr B}$-free sets

Key Result

Theorem B1'

Suppose that $W$ is measurable, relatively compact and Haar aperiodic. Then $({\mathcal{M}}^{ G}_{ W},Q_{ W}^{ G},S)$ is measure-theoretically isomorphic to $({\widetilde{X}},m_{\widetilde{X}}, \widetilde{T})$.

Theorems & Definitions (40)

  • Theorem B1'
  • Theorem B2'
  • Remark 1.1: diffraction analysis
  • Remark 1.2: examples
  • Definition 2.1: Mirsky $k$-genericity
  • Remark 2.2: sets of Mirsky genericity
  • Proposition 2.3: Moody's uniform distribution theorem
  • Remark 2.4: when Mirsky 1-genericity implies Mirsky genericity
  • Remark 2.5: Mirsky genericity on $G$ versus $G \times H$
  • Lemma 2.6
  • ...and 30 more