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

TL;DR

This paper explores the topological and dynamical properties of $k$-free numbers in cyclotomic fields, extending previous quadratic field results to a broader class of algebraic number fields.

Contribution

It extends the analysis of $k$-free integers to cyclotomic fields, examining their entropy and symmetries, and explores the connection between dynamical systems and number theory.

Findings

Analysis of entropy in cyclotomic $k$-free systems

Identification of extended symmetries in these systems

Insights into the interplay between dynamical and number-theoretic properties

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.