S. O. Klymchuk

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Articles: 1

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

M. V. Pratsiovytyi, S. O. Klymchuk, O. P. Makarchuk

MFAT 22 (2016), no. 2, 184-196

184-196

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 $\nu_j(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3.$

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S. O. Klymchuk

Level sets of asymptotic mean of digits function for 44-adic representation of real number

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