Topological, metric and fractal properties of the set of real numbers with a given asymptotic mean of digits in their $4$-adic representation in the case when the digit frequencies exist

Abstract

In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}\alpha_k(x), \text{ where } x=\sum^{\infty}_{k=1}\alpha_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of real number $x$, particularly properties of it's level sets $ S_{\theta}=\left\{x: r(x)=\theta,\: \theta=const, \: 0\leqslant\theta\leqslant 3\right\}, $ if all $4$-adic digits frequencies exist, i.e. $$ \nu_i(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=i, i\leqslant n\}, \:\: i=0,1,2,3. $$ We provided an algorithm of constructing point from the set $S_{\theta}$, and proved continuality and every where density of the set. We found conditions of zero and full Lebesgue measure and estimates of Hausdorff-Besicovitch fractal dimension.