On fractal properties of non-normal numbers with respect to Rényi f -expansions generated by piecewise linear functions

On new fractal phenomena connected with infinite linear IFS

“…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 ‐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.…”

“…Proof The main idea of the proof is rather clear: to construct a countable family of subsets from whose Hausdorff dimension can be arbitrarily close to unity. Possible infinite entropy of the stochastic vector , possible non‐faithfulness of the family and the absence of general formulae for the calculation of the Hausdorff dimension for probability measures with independent ‐symbols (this is still an open problem (see, e.g., )) do not allow us to apply methods from , to construct such a family. Methods from , are also not applicable to solve the problem because of the absence of “divergent points techniques” for the measures generated by infinite IFS.…”

“…Despite the above mentioned difficulties, in the last section of the present work we develop a new approach to the study of subsets of ‐essentially non‐normal numbers and prove (without any additional restrictions on the stochastic vector like convergence of the series (see, e.g., ), finiteness of the entropy or faithfulness of the corresponding family ) that this set is superfractal. 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 of their digits in the generalized Lüroth expansion:…”

On fractal faithfulness and fine fractal properties of random variables with independent $Q^*$-digits

“…and of pure singularly continuous type if and only if infinite products (5) and (6) are equal to zero. Let us recall that the Hausdorff dimension of the distribution of a random variable τ is defined as follows:…”

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

“…The fractal and multifractal approaches to the study of such measures are known to be extremely useful (see, e.g., [7,12,39] and references therein). The study of fractal properties of different families of singularly continuous probability measures (see, e.g., [7,16,26,28,27,29,31,38,1,42] and references therein) can be used to solve non-trivial problems in the metric number theory ( [8,9,5,4,10,30,21]), in the theory of dynamical systems and DP-transformations and in fractal analysis ( [6,3,11,19,18,17,22,41,43]).…”