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arXiv · 2207.05396

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

Aurelia Dymek·Stanisław Kasjan·Joanna Kułaga-Przymus
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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, the $\mathfrak{B}$-free system is minimal if and only if the characteristic function of $\mathfrak{B}$-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 periodic 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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Aurelia Dymek, Stanisław Kasjan, Joanna Kułaga-Przymus. 2022-07-12. Minimality of $\mathfrak{B}$-free systems in number fields. https://arxiv.org/abs/2207.05396

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