On the Hausdorff dimension faithfulness connected with $Q_{infty}$-expansion

TL;DR

This paper proves that certain unions of cylinders in $Q_{\infty}$-expansion are faithful for Hausdorff dimension calculation and provides conditions for the entire family of cylinders to be faithful, resolving an open problem.

Contribution

It establishes the faithfulness of unions of finite consecutive cylinders in $Q_{\infty}$-expansion and characterizes when all cylinders are faithful for Hausdorff dimension.

Findings

Unions of finite consecutive cylinders are faithful for Hausdorff dimension.

Necessary and sufficient conditions for all cylinders to be faithful.

Resolved an open problem on $Q_{\infty}$-expansion faithfulness.

Abstract

In this paper, we show that, the family of all possible union of finite consecutive cylinders of the same rank of $Q_{\infty}$-expansion is faithful for the Hausdorff dimension calculation. Applying this result, we give the necessary and sufficient condition for the family of all cylinders of $Q_{\infty}$-expansion to be faithful for Hausdorff dimension calculation on the unit interval, this answers the open problem mentioned in a paper of S. Albeverio et al..