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

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, $$r(x)=\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{i=1}^n{\alpha }_i(x)$$ if the asymptotic frequency ${\nu }_j(x)$ of at least one digit does not exist, were $$ ν_j(x)=\lim_{n\to\infty}n^{-1}#\{k: α_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$

Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=849