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- Title
Minimality of $ \mathfrak{B} $-free systems in number fields.
- Authors
Dymek, Aurelia; Kasjan, Stanisław; Kułaga-Przymus, Joanna
- Abstract
Let $ K $ be a finite extension of $ \mathbb{Q} $ and $ \mathcal{O}_K $ be its ring of integers. Let $ \mathfrak{B} $ be a primitive collection of ideals in $ \mathcal{O}_K $. We show that any $ \mathfrak{B} $-free system is essentially minimal. Moreoever, a $ \mathfrak{B} $-free system is minimal if and only if the characteristic function of the $ \mathfrak{B} $-free numbers is a Toeplitz sequence. Equivalently, there are no ideal $ \mathfrak{d} $ and no infinite pairwise coprime collection of ideals $ \mathcal{C} $ such that $ \mathfrak{d} \mathcal{C}\subseteq\mathfrak{B} $. Moreover, we find a period structure in the Toeplitz case. Last but not least, we describe the restrictions on the cosets of ideals contained in unions of ideals.
- Subjects
NUMBER systems; RINGS of integers; CHARACTERISTIC functions
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2023, Vol 43, Issue 9, pN.PAG
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2023056