On fractal faithfulness and fine fractal properties of random variables with independent \(Q^*\)-digits (Q340822)

The Hausdorff measure and the Hausdorff dimension of set a \(E\) is defined using arbitrary coverings of \(E\). If one restricts to a smaller family of coverings, then one obtains variants of the Hausdorff measure and dimension. If, when considering a family \(\Phi\) of coverings, every set \(E\) has the same Hausdorff dimension as in the classical case (using arbitrary coverings), then the family \(\Phi\) is called faithful.NEWLINENEWLINEIn this paper the authors, consider coverings generated by cylinders of \(Q^*\)-expansions, a concept which unfortunately is not defined in the paper. The main result of the paper is a theorem giving sufficient conditions for the family of coverings by cylinders of \(Q^*\)-expansions to be faithful.NEWLINENEWLINEAs an application, they study fractal properties of some random variables.