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.
Full Text
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},
}