On fractal faithfulness and fine fractal properties of random variables with independent $\boldsymbol{Q^*}$-digits

TL;DR

This paper introduces a new technique to establish the faithfulness of Hausdorff--Besicovitch dimension calculations for families generated by $Q^*$-expansions, removing previous restrictions and exploring fractal properties of related random variables.

Contribution

It develops a novel method to prove faithfulness of dimension calculations for $Q^*$-expansion families, eliminating technical restrictions and analyzing fractal properties of associated random variables.

Findings

New technique proves faithfulness without restrictions.

All previous restrictions are shown to be technical and removable.

Studied fine fractal properties of random variables with independent $Q^*$-digits.

Abstract

We develop a new technique to prove the faithfulness of the Hausdorff--Besicovitch dimension calculation of the family $\varPhi(Q^*)$ of cylinders generated by $Q^*$-expansion of real numbers. All known sufficient conditions for the family $\varPhi(Q^*)$ to be faithful for the Hausdorff--Besicovitch dimension calculation use different restrictions on entries $q_{0k}$ and $q_{(s-1)k}$. We show that these restrictions are of purely technical nature and can be removed. Based on these new results, we study fine fractal properties of random variables with independent $Q^*$-digits.