Lochs-type theorems beyond positive entropy
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 continue...
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Zusammenfassung: | 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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DOI: | 10.48550/arxiv.2202.04008 |