正熵以外的Lochs型定理
Lochs-type theorems beyond positive entropy
摘要：
洛赫定理及其推广是一种转换定理，它将一个实数的一种展开式中确定的位数与另一种展开式中给定的位数的函数联系起来。在其原始版本中，洛克斯定理将十进制展开式与连分数展开式联系起来。这样的转换结果也可以在适当的假设下对区间划分的序列进行陈述，结果几乎在几乎处处都成立，或者在度量上，涉及熵。这就是我们在这里发展的观点。为了处理正熵以上的分拆序列，本文引入了对数平衡分拆序列的概念，并给出了对数平衡分拆序列的权函数。这些是间隔分区的序列，使得它们在每个深度处的间隔的测量的斜率大致相同。然后，我们国家洛克斯型定理的工作，即使在零熵的情况下，特别是几个重要的日志平衡序列的分区数论性质。
英文摘要：
Lochs' theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. In its original version, Lochs' theorem related decimal expansions with continued fraction expansions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. This is the viewpoint we develop here. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature.
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