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

TL;DR

This paper investigates the minimality properties of $rak{B}$-free systems in number fields, establishing conditions under which these systems are minimal and characterizing their structure via Toeplitz sequences.

Contribution

It provides a characterization of minimal $rak{B}$-free systems in number fields and links their minimality to Toeplitz sequences, extending previous results to algebraic number theory.

Findings

Any $rak{B}$-free system is essentially minimal.

The system is minimal iff the characteristic function is a Toeplitz sequence.

A periodic structure exists in the Toeplitz case.

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.