Symmetries of power-free integers in number fields and their shift spaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909630698881024 |
|---|---|
| author | Gundlach, Fabian Klüners, Jürgen |
| author_facet | Gundlach, Fabian Klüners, Jürgen |
| contents | We describe the group of $\mathbb Z$-linear automorphisms of the ring of integers of a number field $K$ that preserve the set $V_{K,k}$ of $k$th power-free integers: every such map is the composition of a field automorphism and the multiplication by a unit.
We show that those maps together with translations generate the extended symmetry group of the shift space $\mathbb D_{K,k}$ associated to $V_{K,k}$. Moreover, we show that no two such dynamical systems $\mathbb D_{K,k}$ and $\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system of another.
We generalize the concept of $k$th power-free integers to sieves and study the resulting admissible shift spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetries of power-free integers in number fields and their shift spaces Gundlach, Fabian Klüners, Jürgen Number Theory Dynamical Systems 37B10, 11R04, 15A86 We describe the group of $\mathbb Z$-linear automorphisms of the ring of integers of a number field $K$ that preserve the set $V_{K,k}$ of $k$th power-free integers: every such map is the composition of a field automorphism and the multiplication by a unit. We show that those maps together with translations generate the extended symmetry group of the shift space $\mathbb D_{K,k}$ associated to $V_{K,k}$. Moreover, we show that no two such dynamical systems $\mathbb D_{K,k}$ and $\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system of another. We generalize the concept of $k$th power-free integers to sieves and study the resulting admissible shift spaces. |
| title | Symmetries of power-free integers in number fields and their shift spaces |
| topic | Number Theory Dynamical Systems 37B10, 11R04, 15A86 |
| url | https://arxiv.org/abs/2407.08438 |