Pillai关于连分数正态性定理的一个类似及其在连续性中的应用
An analogue of Pillai's theorem for continued fraction normality and an application to subsequences
摘要：
我们表明，两个概念的连分数正常，一个有限模式的重叠发生被视为不同的发生，另一个只有不相交的发生被视为不同的，是相同的。这个等价涉及到一个类似于S的定理。S. Pillai在1940年为基地- $b$扩展。证明需要的技术是根本不同的，因为连分数扩展利用了可数无限字母表，导致一个非紧空间。利用这两个概念的等价性，我们提供了一个新的证明Heersink和Vandehey的最近的结果，沿等差数列的连续性的选择不保持连分数正常。
英文摘要：
We show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This equivalence involves an analogue of a theorem due to S. S. Pillai in 1940 for base-$b$ expansions. The proof requires techniques which are fundamentally different, since the continued fraction expansion utilizes a countably infinite alphabet, leading to a non-compact space. Utilizing the equivalence of these two notions, we provide a new proof of Heersink and Vandehey's recent result that selection of subsequences along arithmetic progressions does not preserve continued fraction normality.
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