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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
MilestonesReceived 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},
}


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