We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non‐faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non‐faithfullness of the family of cylinders generated by the classical Lüroth expansion. We also develop new approach to the study of subsets of Q∞‐essentially non‐normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Lüroth expansion.

We study an equivalent definition of the Hausdorff-Besicovitch dimension in terms of a system Φ(Q ∞ ) of cylinders of the Q ∞ -expansion. Sufficient conditions for the system Φ(Q ∞ ) to be faithful for the evaluation of the Hausdorff-Besicovitch dimension in the unit interval are found; fine fractal properties of probability measures with independent Q ∞ -digits are investigated (we do not assume that the Q ∞digits are identically distributed). k s=1 q i s 2010 Mathematics Subject Classification. Primary 60G30, 11K55, 28A80. Key words and phrases. Q ∞ -expansions, faithful systems of coverings, singularly continuous probability distributions, Hausdorff-Besicovitch dimension of a set, Hausdorff dimension of a measure. The first author was supported by the Project DFG 436113/97. The second author was supported by the Projects DFG 436 UKR 113/97 and DFG KO 1989/6-1 and the Alexander von Humboldt Foundation.

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