Minimality of $ \mathfrak{B} $-free systems in number fields

Aurelia Dymek; Stanisław Kasjan; Joanna Kułaga-Przymus

doi:10.3934/dcds.2023056

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2023, Volume 43, Issue 9: 3512-3548. Doi: 10.3934/dcds.2023056

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Minimality of $ \mathfrak{B} $-free systems in number fields

Nicolaus Copernicus University, Poland

Received: September 2022

Revised: March 2023

Early access: May 2023

Published: September 2023

Research supported by Narodowe Centrum Nauki UMO-2019/33/B/ST1/00364

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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.

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Mathematics Subject Classification: Primary: 37B10.

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Table 1. Results in higher dimension vs. in dimension one

	$ \mathcal{O}_K $	$ {\mathbb{Z}} $
Essential minimality	Theorem A	Theorem 3.3
Minimality characterization	Theorem B	Theorem 3.7
Essence of being Toeplitz	Theorem C	Theorem 3.9
Non-periodic positions on $ \eta $	Proposition D	Proposition 3.16
Period structure	Theorem E	Theorem 3.17
APs in "Toeplitz sets of multiples"	Theorem F	Theorem 3.14

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