Topological and fractal properties of real numbers which are not normal

Bulletin des Sciences Mathématiques - Tập 129 - Trang 615-630 - 2005
Sergio Albeverio1,2,3,4, Mykola Pratsiovytyi5, Grygoriy Torbin5,6
1Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany
2SFB 611, Bonn, BiBoS, Bielefeld, Bonn, Germany
3CERFIM, Locarno and Acc. Arch., USI, Switzerland
4IZKS, Bonn, Germany
5National Pedagogical University, Pyrogova str. 9, 01030 Kyiv, Ukraine
6Institute for Mathematics, NAS of Ukraine, Tereshchenkivs'ka str. 3, 01030, Kyiv, Ukraine

Tài liệu tham khảo

Albeverio, 2004, Fractal probability distributions and transformations preserving the Hausdorff–Besicovitch dimension, Ergodic Theory and Dynamical Systems, 24, 1, 10.1017/S0143385703000397 S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, G. Torbin, Q˜-representation of real numbers and fractal probability distributions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, submitted for publication (SFB 611 Preprint, Universität Bonn) Albeverio, 2005, Fractal properties of singular continuous probability distributions with independent Q∗-digits, Bull. Sci. Math., 129, 356, 10.1016/j.bulsci.2004.12.001 Besicovitch, 1934, On the sum of digits of real numbers represented in the dyadic systems, Math. Ann., 110, 321, 10.1007/BF01448030 Billingsley, 1965 Billingsley, 1961, Hausdorff dimension in probability theory II, Illinois. J. Math., 5, 291, 10.1215/ijm/1255629826 Chatterji, 1964, Certain induced measures and the fractional dimensions of their “supports”, Z. Wahr., 3, 184, 10.1007/BF00534907 Eggleston, 1949, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser., 20, 31, 10.1093/qmath/os-20.1.31 Falconer, 1995 Jessen, 1935, Distribution function and Riemann zeta-function, Trans. Amer. Math. Soc., 38, 48, 10.1090/S0002-9947-1935-1501802-5 Olsen, 2004, Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion, Bull. Sci. Math., 128, 265, 10.1016/j.bulsci.2004.01.003 Olsen, 2004, Applications of multifractal divergence points to sets of numbers defined by their N-adic expansion, Math. Proc. Cambridge Philos. Soc., 136, 139, 10.1017/S0305004103007047 Olsen, 2003, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc. (2), 67, 103, 10.1112/S0024610702003630 Pratsiovytyi, 1998 Pratsiovytyi, 1995, Superfractality of the set of numbers having no frequency of n-adic digits, and fractal probability distributions, Ukrainian Math. J., 47, 971 Schweiger, 1995 Turbin, 1992