Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits
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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.
KeywordsProbability Distribution Real Number Fractal Property Unit Interval Fractal Geometry
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