Topological and fractal properties of real numbers which are not normal

Abstract

The set L of essentially non-normal numbers of the unit interval (i.e., the set of real numbers having no asymptotic frequencies of all digits in their nonterminating s-adic expansion) is studied in details. It is proven that the set L is generic in the topological sense (it is of the second Baire category) as well as in the sense of fractal geometry ( L is a superfractal set, i.e., the Hausdorff–Besicovitch dimension of the set L is equal 1). These results are substantial generalizations of the previous results of the two latter authors [M. Pratsiovytyi, G. Torbin, Ukrainian Math. J. 47 (7) (1995) 971–975]. The Q ∗ -representation of real numbers (which is a generalization of the s-adic expansion) is also studied. This representation is determined by the stochastic matrix Q ∗ . We prove the existence of such a Q ∗ -representation that almost all (in the sense of Lebesgue measure) real numbers have no asymptotic frequency of all digits. In the case where the matrix Q ∗ has additional asymptotic properties, the Hausdorff–Besicovitch dimension of the set of numbers with prescribed asymptotic properties of their digits is determined (this is a generalization of the Eggleston–Besicovitch theorem). The connections between the notions of “normality of numbers” respectively of “asymptotic frequencies” of their digits is also studied.