On $k$-free numbers in cyclotomic fields: entropy, symmetries and topological invariants
Volume 281 / 2025
Studia Mathematica 281 (2025), 275-301
MSC: Primary 37B10; Secondary 52C23
DOI: 10.4064/sm240212-15-2
Published online: 17 April 2025
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 $\mathcal B$-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.
