On the Hausdorff dimension faithfulness and the Cantor series expansion

We study families \Phi of coverings which are faithful for the Hausdorff
dimension calculation on a given set E (i. e., special relatively narrow families of
coverings leading to the classical Hausdorff dimension of an arbitrary subset of E) and
which are natural generalizations of comparable net-coverings. They are shown to be
very useful for the determination or estimation of the Hausdorff dimension of sets and
probability measures. We give general necessary and sufficient conditions for a covering
family to be faithful and new techniques for proving faithfulness/non-faithfulness
for the family of cylinders generated by expansions of real numbers. Motivated by
applications in the multifractal analysis of infinite Bernoulli convolutions, we study
in details the Cantor series expansion and prove necessary and sufficient conditions
for the corresponding net-coverings to be faithful. To the best of our knowledge this
is the first known sharp condition of the faithfulness for a class of covering families
containing both faithful and non-faithful ones. Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families \scrA of net-coverings which are "extremely non-comparable" to the Hausdorff measure.