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Level sets of asymptotic mean of digits function for $4$-adic representation of real number


Abstract

We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}\alpha_i(x)$$ if the asymptotic frequency $\nu_j(x)$ of at least one digit does not exist, were $$ \nu_j(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$

Key words: $s$-Adic representation, asymptotic mean of digits function, level sets, frequency of digits, Besicovitch-Eggleston sets, Hausdorff-Besicovitch dimension.


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Article Information

TitleLevel sets of asymptotic mean of digits function for $4$-adic representation of real number
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 184-196
MathSciNet   MR3522859
zbMATH 06665387
Milestones  Received 04/09/2014; Revised 20/10/2015
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

M. V. Pratsiovytyi
National Pedagogical Dragomanov University, 9 Pirogova, Kyiv, 01601, Ukraine; Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

S. O. Klymchuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

O. P. Makarchuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine


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Citation Example

M. V. Pratsiovytyi, S. O. Klymchuk, and O. P. Makarchuk, Level sets of asymptotic mean of digits function for $4$-adic representation of real number, Methods Funct. Anal. Topology 22 (2016), no. 2, 184-196.


BibTex

@article {MFAT849,
    AUTHOR = {Pratsiovytyi, M. V. and Klymchuk, S. O. and Makarchuk, O. P.},
     TITLE = {Level sets of asymptotic mean of digits function for $4$-adic representation of real number},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {2},
     PAGES = {184-196},
      ISSN = {1029-3531},
  MRNUMBER = {MR3522859},
 ZBLNUMBER = {06665387},
       URL = {http://mfat.imath.kiev.ua/article/?id=849},
}


References

  1. S. Albeverio, M. Pratsiovytyi, and G. Torbin, Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their $s$-adic digits, Ukrainian Math. J. 57 (2005), no. 9, 1361-1370.  MathSciNet CrossRef
  2. Sergio Albeverio, Mykola Pratsiovytyi, and Grygoriy Torbin, Topological and fractal properties of real numbers which are not normal, Bull. Sci. Math. 129 (2005), no. 8, 615-630.  MathSciNet CrossRef
  3. A. S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1935), no. 1, 321-330.  MathSciNet CrossRef
  4. Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965.  MathSciNet
  5. E. Borel, Les probabilites denombrables et leurs applications arithmetiques, Rend. Circ. Mat. 27 (1909), no. 1, 247-271.
  6. H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math., Oxford Ser. 20 (1949), 31-36.  MathSciNet
  7. 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. 67 (2003), no. 1, 103-122.  MathSciNet CrossRef
  8. M. V. Pratsiovytyi, S. O. Klymchuk, Linear fractals of Besicovitch-Eggleston type, Scientific journal of Dragomanov NPU. Series 1. Physics and mathematics (2012), no. 13(2), 80-92. (Ukrainian)
  9. M. V. Pratsiovytyi, S. O. Klymchuk, Topological, metric and fractal properties of sets of real numbers with preassigned mean of digits of 4-adic representation when their frequencies exist, Scientific journal of Dragomanov NPU. Series 1. Physics and mathematics (2013), no. 14, 217-226. (Ukrainian)
  10. S. O. Klymchuk, O. P. Makarchuk, and M. V. Pratsovytyi, Frequency of a digit in the representation of a number and the asymptotic mean value of the digits, Ukrainian Math. J. 66 (2014), no. 3, 336-346.  MathSciNet CrossRef
  11. M. V. Prats′ovitii and G. M. Torbin, Superfractality of the set of numbers having no frequency of $n$-adic digits, and fractal distributions of probabilities, Ukrainian Math. J. 47 (1995), no. 7, 1113-1118.  MathSciNet CrossRef
  12. G. M. Torbin, Frequency characteristics of normal numbers in different number systems, Fractal Analysis and Related Problems, Kyiv: Institute of Mathematics, National Academy of Sciences of Ukraine - National Pedagogical Dragomanov University, 1998, no. 1, pp. 53-55. (Ukrainian)


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