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

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..

arXiv comment10 pages