康托级数构造与两种正规性概念的比较
Cantor Series Constructions Contrasting Two Notions of Normality
摘要：
A. Rényi \cite{Renyi}在$Q$ -Cantor级数的背景下给出了一个简单正规性的推广定义。在\cite{Mance}中，给出了$Q$ -正规性的定义，它推广了$Q$ -Cantor级数中的正规性概念。在这项工作中，我们检查$Q$ -正态性和$Q$ -分布正态性，在\cite{Laffer}和\cite{Salat}处理。具体来说，虽然这两个概念的非等价性是隐含在\cite{Laffer}，在本文中，我们给出了一个明确的建设见证非平凡的方向。也就是说，我们构建了一个基本的$Q$以及一个真正的$x$，它是$Q$ -正态的，但不是$Q$ -分布正态的。接下来，我们通过构造一个基$Q$和一个既为$Q$ -正态分布又为$Q$ -分布正态分布的真实$x$的显式例子来探讨同时正态性的问题。
英文摘要：
A. Rényi \cite{Renyi} made a definition that gives a generalization of simple normality in the context of $Q$-Cantor series. In \cite{Mance}, a definition of $Q$-normality was given that generalizes the notion of normality in the context of $Q$-Cantor series. In this work, we examine both $Q$-normality and $Q$-distribution normality, treated in \cite{Laffer} and \cite{Salat}. Specifically, while the non-equivalence of these two notions is implicit in \cite{Laffer}, in this paper, we give an explicit construction witnessing the nontrivial direction. That is, we construct a base $Q$ as well as a real $x$ that is $Q$-normal yet not $Q$-distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base $Q$ as well as a real $x$ that is both $Q$-normal and $Q$-distribution normal.
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