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

TL;DR

This paper investigates the topological, metric, and fractal characteristics of level sets of the asymptotic mean of digits function in 4-adic representations of real numbers, revealing complex structural properties.

Contribution

It provides a detailed analysis of the properties of level sets of the asymptotic mean of digits function in 4-adic expansions, including cases with non-existent digit frequency limits.

Findings

Characterization of level sets' fractal dimensions

Description of topological structure of level sets

Analysis of metric properties and measure-theoretic aspects

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