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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.

### Article Information

 Title Level sets of asymptotic mean of digits function for $4$-adic representation of real number Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 184-196 MathSciNet MR3522859 zbMATH 06665387 Milestones Received 04/09/2014; Revised 20/10/2015 Copyright The 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

### 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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