Superfractal

As a corollary of our main results, we obtain the non-faithfullness of the family of cylinders generated by the classical L\"uroth expansion. Possible infinite entropy of the stochastic vector Q#95;#92;infty which determines the metric relations for partitions of the generalized L\"uroth expansions, possible non-faithfulness of the family #92;Phi(Q#95;#92;infty) and the absence of general formulae for the calculation of the Hausdorff dimension even for probability measures with independent identically distributed symbols of generalized L\"uroth expansions are all facts that do not allow to apply usual probabilistic methods, as well as methods of dynamical systems to study fractal properties of corresponding subsets of non-normal numbers. Despite of the above mentioned difficulties, we develop new approach to the study of subsets of Q#95;#92;infty-essentially non-normal numbers and prove (without any additional restrictions on the stochastic vector Q#95;#123;#92;infty#125;) that this set is superfractal.