Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits

Properties of the set T s of "particularly nonnormal numbers" of the unit interval are studied in detail (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 proved that the set T s is residual in the topological sense (i.e., it is of the first Baire category) and 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. © 2005 Springer Science+Business Media, Inc.