Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits - Pratsiovytyi, M. V.; Torbin, H. M. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
Dedicated to V. S. Korolyuk on occasion of his 80-th birthday Properties of the set T_s of "particularly nonnormal numbers" of the unit interval are studied in details (T_s consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proven that the set T_s is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ( T_s is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented.