无$\mathcal B$系统的广义遗传
Generalized heredity in $\mathcal B$-free systems
摘要：
设$\mathcal B\subseteq\mathbb N$是一个本原集。我们在两个方向上补充了[arxiv：1509.08010]中关于无$\mathcal B$的子位移$X_η$的遗传性的结果：在邻近的情况下，我们证明了可能比子位移$X_η$稍大的子位移$X_φ$总是遗传的。（不需要假设集合$\mathcal B$是拉紧的或者甚至具有轻尾，但是如果$\mathcal B$是拉紧的，则$X_φ=X_η$。）我们还推广了遗传的概念，以非邻近（因此非遗传）的情况下，证明$X_φ$总是“遗传远离其唯一的最小子系统”（这总是Toeplitz）。最后，我们等价地用条件$m_H(\overline{\operatorname{int}(W)})=0$刻画了Toeplitz子系统的正则性，其中$W$（“窗口”）是紧交换群$H$的子集，它与集合$\mathcal B$正则相关，$m_H$表示$H$上的Haar测度.在整个过程中，[arxiv：1702.02375]的结果被大量使用。
英文摘要：
Let $\mathcal B\subseteq\mathbb N$ be a primitive set. We complement results on heredity of the $\mathcal B$-free subshift $X_η$ from [arxiv:1509.08010] in two directions: In the proximal case we prove that a subshift $X_φ$, which micht be slightly larger than the subshift $X_η$, is always hereditary. (There is no need to assume that the set $\mathcal B$ is taut or even has light tails, but if $\mathcal B$ is taut, then $X_φ=X_η$.) We also generalize the the concept of heredity to the non-proximal (and hence non-hereditary) case by proving that $X_φ$ is always "hereditary away from its unique minimal subsystem" (which is always Toeplitz). Finally we characterize regularity of this Toeplitz subsystem equivalently by the condition $m_H(\overline{\operatorname{int}(W)})=0$, where $W$ ("the window") is a subset of a compact abelian group $H$ canonically associated with the set $\mathcal B$, and $m_H$ denotes Haar measure on $H$. Throughout, results from [arxiv:1702.02375] are heavily used.
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