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On $k$-free numbers in cyclotomic fields: entropy, symmetries and topological invariants

Michael Baake, Alvaro Bustos, Andreas Nickel

TL;DR

This work extends the study of $k$-free integers from quadratic and related number fields to cyclotomic fields, formulating the problem within a cut-and-project framework and treating the resulting point sets as weak model sets with maximal density. It establishes a precise entropy–density connection, showing $h_{ ext{top}} = h_{ ext{pc}} = \log(2)/\zeta_K(k)$, and proves a pure-point dynamical spectrum under the Mirsky measure, with the measure of maximal entropy carrying Lebesgue components. The stabiliser of the cyclotomic $k$-free set is shown to be the semidirect product $\\mathcal{O}_n^{\\times} \\rtimes \\mathrm{Gal}(K/\\mathbb{Q})$, and the extended symmetries (normaliser) are given by $\\mathcal{R} = \\mathcal{S} \\rtimes \\mathcal{E}$ with $\\mathcal{E} \\simeq \\mathcal{O}_n^{\\times} \\rtimes \\mathrm{Aut}_{\\mathbb{Q}}(K)$. These results link intrinsic number-theoretic data (unit groups, Galois groups, Dedekind zeta values) to dynamical invariants and symmetry structures, and they pave the way for analogous analyses in broader Galois or non-Galois algebraic settings.

Abstract

Point sets of number-theoretic origin, such as the visible lattice points or the $k$-th power free integers, have interesting geometric and spectral properties and give rise to topological dynamical systems that belong to a large class of subshifts with positive topological entropy. Among them are $\cB$-free systems in one dimension and their higher-dimensional generalisations, most prominently the $k$-free integers in algebraic number fields. Here, we extend previous work on quadratic fields to the class of cyclotomic fields. In particular, we discuss their entropy and extended symmetries, with special focus on the interplay between dynamical and number-theoretic notions.

On $k$-free numbers in cyclotomic fields: entropy, symmetries and topological invariants

TL;DR

This work extends the study of -free integers from quadratic and related number fields to cyclotomic fields, formulating the problem within a cut-and-project framework and treating the resulting point sets as weak model sets with maximal density. It establishes a precise entropy–density connection, showing , and proves a pure-point dynamical spectrum under the Mirsky measure, with the measure of maximal entropy carrying Lebesgue components. The stabiliser of the cyclotomic -free set is shown to be the semidirect product , and the extended symmetries (normaliser) are given by with . These results link intrinsic number-theoretic data (unit groups, Galois groups, Dedekind zeta values) to dynamical invariants and symmetry structures, and they pave the way for analogous analyses in broader Galois or non-Galois algebraic settings.

Abstract

Point sets of number-theoretic origin, such as the visible lattice points or the -th power free integers, have interesting geometric and spectral properties and give rise to topological dynamical systems that belong to a large class of subshifts with positive topological entropy. Among them are -free systems in one dimension and their higher-dimensional generalisations, most prominently the -free integers in algebraic number fields. Here, we extend previous work on quadratic fields to the class of cyclotomic fields. In particular, we discuss their entropy and extended symmetries, with special focus on the interplay between dynamical and number-theoretic notions.
Paper Structure (6 sections, 16 theorems, 53 equations)

This paper contains 6 sections, 16 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and known results
  3. Spectral structure and topological entropy
  4. Intermezzo: Algebraic number fields, units, and ramified primes
  5. The stabiliser of $k$-free integers in cyclotomic fields
  6. Extended symmetries and further invariants

Key Result

Lemma 3.2

The patch counting entropy of $V'_k$ exists and is given by $h_{\mathrm{pc}} = \log (2) / \zeta^{}_{K} (k)$.

Theorems & Definitions (33)

  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • proof
  • Theorem 3.6
  • ...and 23 more