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Article Dans Une Revue Monatshefte für Mathematik Année : 2022

LOCHS-TYPE THEOREMS BEYOND POSITIVE ENTROPY

Eda Cesaratto
Pablo Rotondo
Martín Safe

Résumé

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 logbalanced sequences of partitions of a number-theoretic nature.
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Dates et versions

hal-04295723 , version 1 (20-11-2023)

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Valérie Berthé, Eda Cesaratto, Pablo Rotondo, Martín Safe. LOCHS-TYPE THEOREMS BEYOND POSITIVE ENTROPY. Monatshefte für Mathematik, 2022, 200 (4), pp.737-779. ⟨10.1007/s00605-022-01805-y⟩. ⟨hal-04295723⟩
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