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

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.

Locations

• Discrete and Continuous Dynamical Systems - View - PDF
• arXiv (Cornell University) - View - PDF