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Dynamical spectrum of power-free integers in quadratic number fields and beyond

Michael Baake, Daniel Luz, Tanja Schindler

TL;DR

The paper develops a unified spectral framework for power-free and $\mathcal{B}$-free lattice systems, using Mirsky measures and the Halmos–von Neumann theorem to obtain explicit pure-point dynamical spectra and continuous eigenfunctions on large measure sets. It then extends the approach to power-free integers in quadratic number fields by embedding rings of integers into cut-and-project schemes, deriving explicit Fourier–Bohr coefficients and the Fourier–Bohr spectrum $L^{\circledast}$ via dual lattices. The results cover both imaginary and real quadratic fields, with concrete treatments of Gaussian and Eisenstein integers and general cases, highlighting how different fields yield distinct spectral invariants and non-conjugate dynamical systems. The work links number theory, ergodic theory, and diffraction theory, providing explicit spectral invariants and density formulas that distinguish measure-theoretic dynamics across arithmetic contexts.

Abstract

Power-free integers and related lattice subsets give rise to interesting dynamical systems. They are revisited from a spectral perspective, in the setting of the Halmos--von Neumann theorem. With respect to the natural patch frequency measure, also known as the Mirsky measure, many of these systems have pure-point dynamical spectrum, but trivial topological point spectrum. We calculate the spectra explicitly, in additive notation, and derive their group structure, both for a large class of $\cB$-free lattice systems in $\RR^d$ and for power-free integers in quadratic number fields. Further, in all cases, the eigenfunctions can be given in closed form, via the Fourier--Bohr coefficients of generic elements and their translates, which form a subset of full Mirsky measure. Based on a simple argument via Kolmogorov's strong law of large numbers, we show how the Fourier--Bohr coefficients also provide the eigenfunctions for the unique measure of maximal entropy, and that we get phase consistency for both measures.

Dynamical spectrum of power-free integers in quadratic number fields and beyond

TL;DR

The paper develops a unified spectral framework for power-free and -free lattice systems, using Mirsky measures and the Halmos–von Neumann theorem to obtain explicit pure-point dynamical spectra and continuous eigenfunctions on large measure sets. It then extends the approach to power-free integers in quadratic number fields by embedding rings of integers into cut-and-project schemes, deriving explicit Fourier–Bohr coefficients and the Fourier–Bohr spectrum via dual lattices. The results cover both imaginary and real quadratic fields, with concrete treatments of Gaussian and Eisenstein integers and general cases, highlighting how different fields yield distinct spectral invariants and non-conjugate dynamical systems. The work links number theory, ergodic theory, and diffraction theory, providing explicit spectral invariants and density formulas that distinguish measure-theoretic dynamics across arithmetic contexts.

Abstract

Power-free integers and related lattice subsets give rise to interesting dynamical systems. They are revisited from a spectral perspective, in the setting of the Halmos--von Neumann theorem. With respect to the natural patch frequency measure, also known as the Mirsky measure, many of these systems have pure-point dynamical spectrum, but trivial topological point spectrum. We calculate the spectra explicitly, in additive notation, and derive their group structure, both for a large class of -free lattice systems in and for power-free integers in quadratic number fields. Further, in all cases, the eigenfunctions can be given in closed form, via the Fourier--Bohr coefficients of generic elements and their translates, which form a subset of full Mirsky measure. Based on a simple argument via Kolmogorov's strong law of large numbers, we show how the Fourier--Bohr coefficients also provide the eigenfunctions for the unique measure of maximal entropy, and that we get phase consistency for both measures.

Paper Structure

This paper contains 14 sections, 15 theorems, 108 equations.

Table of Contents

  1. Introduction and overview
  2. A guiding example with many facets
  3. Power-free integers and some of their generalisations
  4. A more general setting with $\mathcal{B}$-free lattice systems
  5. Quadratic number fields
  6. Imaginary quadratic fields
  7. Gaussian integers
  8. Eisenstein integers
  9. General imaginary quadratic field
  10. Real quadratic fields
  11. The case $d=2$
  12. The case $d=5$
  13. General real quadratic field
  14. Bernoulli thinning of uniformly distributed sequences

Key Result

Theorem 1

The dynamical system $(\mathbb{X}^{\space}_{V},\space\mathbb{Z}^d, \mu_{_\mathrm{M}})$ has pure-point spectrum. In additive notation, the spectrum is given by the group $\{ k \in \mathbb{Q}\space^d\cap \mathbb{T}^d : \mathrm{den} (k) \text{ is square-free} \} \subset \mathbb{T}^d$. The corresponding

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof : Sketch of proof
  • Remark 6
  • Corollary 7
  • Theorem 8
  • Remark 9
  • ...and 18 more