动态生成康托尔级数展开的正则性理论
The Theory of Normality for Dynamically Generated Cantor Series Expansions
AI总结：
研究动态生成康托尔级数展开的正则性理论，发现其与基数g展开理论有相似之处，并提出热点定理。
摘要：
实数在[0,1)区间内以基数g展开的正则性理论内容丰富且发展完善。类似理论已发展出许多其他数制系统，如常规连续分数展开、β展开和卢罗系列展开。令Q=(q_n)_{n∈N}为一组大于等于2的整数序列。实数x∈[0,1)的Q-康托尔级数展开是唯一形如x=∑_{n=1}^∞x_n/(q_1q_2⋯q_n)的表达式，其中x_n≠q_n-1无限次出现。对于康托尔级数展开，迄今为止大部分文献考虑的是Q，其正则性理论与基数g展开的理论有显著差异。我们引入了动态生成的康托尔级数展开这一类别，这是一个包含许多例子的康托尔级数展开集合，其中大部分经典基数g展开的理论可以并行发展。该类别包括许多例子，如在{2,3}上的图尔-莫尔斯序列和翻译后的夏普尔奈数。我们主要结果的一个特殊情况是，如果Q是一个有界基本序列，并且由一个具有零熵的保测系统动态生成，那么以Q为基数的正则性与以Q为基数的分布正则性一致，并且Q具有热点定理。
英文摘要：
The theory of normality for base $g$ expansions of real numbers in $[0,1)$ is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion, $β$-expansions, and Lüroth series expansions. Let $Q=(q_n)_{n \in \mathbb{N}}$ be a sequence of integers greater than or equal to 2. The $Q$-Cantor series expansion of $x \in [0,1)$ is the unique sum of the form $x=\sum_{n=1}^\infty \frac{x_n}{q_1q_2\cdots q_n}$, where $x_n \neq q_n-1$ infinitely often. For the Cantor series expansions, most of the literature thus far considers $Q$ where the theory of normality differs drastically from that of the base $g$ expansions. We introduce the class of dynamically generated Cantor series expansions, which is a large class of Cantor series expansions for which much of the classical theory of base $g$ expansions can be developed in parallel. This class includes many examples such as the Thue-Morse sequence on $\{2,3\}$ and translated Champernowne numbers. A special case of our main results is that if $Q$ is a bounded basic sequence that is dynamically generated by an ergodic system having zero entropy, then normality base $Q$ coincides with distribution normality base $Q$, and $Q$ possesses a Hot Spot Theorem.
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