Lochs-type theorems beyond positive entropy

TL;DR

This paper extends Lochs' theorems to zero entropy cases by introducing log-balanced sequences of partitions, enabling conversion results for a broader class of number expansions beyond positive entropy.

Contribution

It introduces log-balanced sequences of partitions and establishes Lochs-type theorems applicable to zero entropy scenarios, broadening the scope of conversion theorems.

Findings

Lochs-type theorems are valid for zero entropy sequences.

Log-balanced sequences have roughly equal logarithmic measures at each depth.

The results apply to number-theoretic partitions beyond positive entropy.

Abstract

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.