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 $$y=r(x)=\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=1}^{\infty }{\alpha }_k(x),\text{ where }x=\sum _{k=1}^{\infty }{\alpha }_k(x)4^{-k}$$ of $4$-adic digits asymptotic mean of fractional part of real number $x$, particularly properties of it's level sets $S_{\theta }=\left\{x:r(x)=\theta ,\theta =const,0⩽\theta ⩽3\right\},$ if all $4$-adic digits frequencies exist, i.e. $${\nu }_i(x)=\underset{n\to \infty }{\lim }{n}^{-1}\#\{k:{\alpha }_k(x)=i,i⩽n\},i=0,1,2,3.$$ 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.