论文标题
湖泊型定理超出正熵
Lochs-type theorems beyond positive entropy
论文作者
论文摘要
Lochs的定理及其概括是转换定理，它们将实际数字的一个扩展中确定的数字数与其他一些扩展中给出的数字数的函数相关联。 Lochs定理与原始版本相关的十进制扩展，并持续扩展。在适当的假设下的间隔分区序列也可以说明这种转换结果，几乎到处或涉及熵的结果。这是我们在这里发展的观点。为了处理以外的正熵之外的分区序列，本文介绍了分区的对数平衡序列的概念，以及其重量函数。这些是间隔分区的序列，使其在每个深度处的间隔度量的对数大致相同。然后，我们指出，即使在零熵的情况下，也可以使用，尤其是对于几个重要的理论分区的几个重要的对数平衡序列，尤其是在零熵的情况下起作用。
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.
