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

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

2006-05-30

arxiv.org/abs/math/0605763v1

Ukrainian Mathematical Journal, 57 (2005), no.9, 1361-1370

Mathematics

Number Theory

Published in Ukrainian Mathematical Journal, 57 (2005), no.9, 1361-1370

Scientific paper

Properties of the set $T_s$ of "particularly non-normal 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.

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