Abstract
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.
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REFERENCES
A. Besicovitch, “On the sum of digits of real numbers represented in the dyadic systems,” Math. Ann., 110, 321–330 (1934).
H. G. Eggleston, “The fractional dimension of a set defined by decimal properties,” Quart. J. Math. Oxford Ser., 20, 31–36 (1949).
M. Prats'ovytyi and G. Torbin, “Superfractality of the set of numbers having no frequency of n-adic digits, and fractal probability distributions,” Ukr. Mat. Zh., 47, No.7, 971–975 (1995).
L. Olsen, “Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion,” Bull. Sci. Math., 128, 265–289 (2004).
L. Olsen, “Applications of multifractal divergence points to sets of numbers defined by their N-adic expansion,” Math. Proc. Cambridge Phil. Soc., 136, No.1, 139–165 (2004).
L. Olsen and S. Winter, “Normal and non-normal points of self-similar sets and divergence points of self-similar measures,” J. London Math. Soc., 2(67), No. 1, 103–122 (2003).
S. Albeverio, M. Pratsiovytyi, and G. Torbin, “Topological and fractal properties of subsets of real numbers which are not normal,” Bull. Sci. Math., No. 208 (2004).
S. Albeverio and G. Torbin, “Fractal properties of singular continuous probability distributions with independent Q*-digits,” Bull. Sci. Math., 129, No.4, 356–367 (2005).
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Dedicated to V. S. Korolyuk on occasion of his 80th birthday
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Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1163–1170, September, 2005.
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Albeverio, S., Prats'ovytyi, M. & Torbin, G. Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits. Ukr Math J 57, 1361–1370 (2005). https://doi.org/10.1007/s11253-006-0001-0
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DOI: https://doi.org/10.1007/s11253-006-0001-0