Harko I. Fractal properties of probability measures generated by polybasic expansions of real numbers, and its applications

Thesis is devoted to the development of methods for the study of singularly continuous probability measures generated by polybasic expansions of real numbers. We develop a new method for the proving of faithfulness for the calculation of the Hausdorff-Besicovitch dimension of systems of coverings on the unit interval. We found sufficient conditions of faithfulness for the calculation of the Hausdorff-Besicovitch dimension of families of overcylindrical sets generated by I-F-representation of real numbers. We investigated the Lebesgue structure, topological-metric properties of spectra of distributions of random variables with independent I-Q_infty-digits. We get the exact formula for the Hausdorff dimension of distributions of random variables with independent I-Q_infty-digits. We proved that a mapping under which symbols of Q_infty-representation are mapped into the same symbols of I-Q_infty-representation is G-isomorphism; based on these results it shown the isomorphism of probabilistic and dimensional theories of Q_infty- and I-Q_infty-representations of real numbers. We investigated the fractal properties of spectra of distribution of random variables with independent I-Q_infty-digits. Using the probabilistic approach we proved that sets of Q- and I-Q_infty-essential non-normal numbers are superfractals. We refuted the conjecture about the superfractality of sets of non-normal and essentially non-normal numbers regardless of choice the system of numeration; sufficient conditions for zero dimensionality of the set of Q*-essentially non-normal numbers are also presented.