论文标题
对正态性保存的Kamae-Weiss定理的确定性功能以及概括
Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation
论文作者
论文摘要
一个古典的kamae-weiss定理指出，较低密度的序列$（n_i）_ {i \ in \ mathbb n} $的正密度为正密度为\ emph {stormality保留}，即具有任何正常的二进制序列$（b_n）_ {n \ in \ mathbb n} $ in \ mathbb n} $的属性$（b_ {n_i}）_ {i \ in \ mathbb n} $是正常的，并且仅当且仅当$（n_i）_ {i \ in \ Mathbb n} $是确定性的序列。给定一个可计数的可计算良好的semigroup $ g $，以及følner序列$ \ nathcal f =（f_n）_ {n \ in \ g $中的\ m athbb n} $，我们介绍了$ g $ utto $ g $的$ g $ severs $ g $ $ \ $ f $ f $ f $ g $ severs的概念的概念f $密度这三个概念是等效的。证据利用了易月群体的瓷砖理论和瓷砖渗透的概念。我们还证明，在$ \ Mathcal f $上的自然假设下，正常保存是正常的，较低$ \ Mathcal f $密度遵循。最后，我们提供了许多在各种半群中保存正态性集的示例
A classical Kamae-Weiss theorem states that an increasing sequence $(n_i)_{i\in\mathbb N}$ of positive lower density is \emph{normality preserving}, i.e. has the property that for any normal binary sequence $(b_n)_{n\in\mathbb N}$, the sequence $(b_{n_i})_{i\in\mathbb N}$ is normal, if and only if $(n_i)_{i\in\mathbb N}$ is a deterministic sequence. Given a countable cancellative amenable semigroup $G$, and a Følner sequence $\mathcal F=(F_n)_{n\in\mathbb N}$ in $G$, we introduce the notions of normality preservation, determinism and subexponential complexity for subsets of $G$ with respect to $\mathcal F$, and show that for sets of positive lower $\mathcal F$-density these three notions are equivalent. The proof utilizes the apparatus of the theory of tilings of amenable groups and the notion of tile-entropy. We also prove that under a natural assumption on $\mathcal F$, positive lower $\mathcal F$-density follows from normality preservation. Finally, we provide numerous examples of normality preserving sets in various semigroups
