On the Hausdorff dimension faithfulness and the Cantor series expansion

TL;DR

This paper investigates conditions under which specific covering families accurately determine Hausdorff dimensions, focusing on the Cantor series expansion and its applications in multifractal analysis.

Contribution

It provides necessary and sufficient conditions for faithfulness of covering families, especially for the Cantor series expansion, and introduces new techniques for analyzing their properties.

Findings

Characterized when net-coverings are faithful for Hausdorff dimension

Established sharp conditions for faithfulness of Cantor series expansions

Constructed examples of faithful coverings that differ from traditional measures

Abstract

We study families $\Phi$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of $E$) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones. Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families $\mathcal{A}$ of net-coverings which are "extremely non-comparable" to the Hausdorff measure.