Level sets of asymptotic mean of digit function for 4-adic representation of real number. (English) Zbl 1363.28016

Pratsiovytï, M. V.; Klimchuk, S. O.; Makarchuk, O. P.

Methods Funct. Anal. Topol. 22, No. 2, 184-196 (2016).

Let \(\alpha_i(x)\), \(i=1,2,\dots\), be the digits of a real number \(x\) in its 4-adic representation. The authors consider the asymptotic mean \[ v(x)=\lim\limits_{n\to \infty}\frac1n \sum\limits_{i=1}^n\alpha_i (x) \] and study topological, metric and fractal properties of the level sets \(S_\theta =\{ x:\;v(x)=\theta\}\) for the case where the asymptotic frequency \(v_j(x)\) of at least one digit does not exist. Here \[ v_j(x)=\lim\limits_{n\to \infty}n^{-1}\# \{ k:\;\alpha_k(x)=j,k\leq n\} \] for \(j=0,1,2,3\).

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

28A80	Fractals
11A63	Radix representation; digital problems
11K16	Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Keywords:

digit function; asymptotic mean; 4-adic representation