On faithfulness and DP-transformations generated by arithmetic Cantor series expansions

TL;DR

This paper investigates the conditions under which families of cylinders generated by Cantor series expansions preserve Hausdorff-Besicovitch dimension, identifying both faithful and non-faithful cases and analyzing dimension preservation under probability distributions.

Contribution

It characterizes when Cantor series expansion cylinders are Hausdorff-Besicovitch faithful and explores dimension preservation by probability distributions of random variables with such expansions.

Findings

Existence of non-faithful subgeometric Cantor series expansions.

Identification of a wide subfamily of faithful Cantor series expansions.

Conditions for dimension preservation under probability distributions.

Abstract

The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions. We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the unit interval. On the other hand we found a rather wide subfamily of subgeometric Cantor series expansions generating faithful families of cylinders. We also study conditions for the Hausdorff-Besicovitch dimension preservation on [0;1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansions.