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

TL;DR

This paper investigates the topological, metric, and fractal properties of the set of real numbers with a specified asymptotic mean of their 4-adic digits, analyzing level sets, measure, and fractal dimensions.

Contribution

It introduces an algorithm for constructing points in the level sets and establishes their continuity, density, measure properties, and fractal dimension estimates.

Findings

The set of points with a given asymptotic mean is everywhere dense.

Conditions for zero and full Lebesgue measure are identified.

Fractal dimension estimates of the level sets are provided.

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.