On the Hausdorff dimension faithfulness connected with   $Q_{infty}$-expansion
  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.. 
Jia Liu and Zhenliang Zhang
math.NT