论文标题
$ \ mathfrak {b} $的最小值 - 数字字段中的免费系统
Minimality of $\mathfrak{B}$-free systems in number fields
论文作者
论文摘要
令$ k $为$ \ mathbb {q} $的有限扩展名，而$ \ mathcal {o} _k $是其整数环。令$ \ mathfrak {b} $为$ \ mathcal {o} _k $中的理想集合。我们证明任何$ \ mathfrak {b} $ - 免费系统本质上是最小的。 Moreo，$ \ mathfrak {b} $ - 自由系统在且仅当$ \ mathfrak的特征函数{b} $ - 免费数字是一个toeplitz序列。同等地，没有理想的$ \ mathfrak {d} $，也没有Infinite成对的coprime集合$ \ mathcal {c} $，因此$ \ mathfrak {d} \ mathcal {c} \ subseteeq \ mathfrak {b} $。此外，我们在Toeplitz情况下找到了一个周期性结构。最后但并非最不重要的一点是，我们描述了对理想工会中所包含的理想的努力的限制。
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.
