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

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 $$j=0,1,2,3$$.

### 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
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