Метрична та ймовірнісна теорії $G_2$-зображення чисел

Abstract

For numbers of interval $[0,g_0]$, $g_0<1$, we consider a
system of encoding with two bases having different signs $g_0$ and
$g_1\equiv g_0-1$ by the means of alphabet $A=\{0,1\}$:
\begin{equation}\label{ex:ab:1}
x=\alpha_1 g_{1-\alpha_1}+\sum\limits^\infty_{k=2}(\alpha_k
g_{1-\alpha_k}\prod\limits^{k-1}_{j=1}g_{\alpha_j})\equiv
\Delta^{G_2}_{\alpha_1\alpha_2\ldots\alpha_k\ldots},
\end{equation}
where $\alpha_n\in A$.
We develop a probabilistic theory for this representation, i.e., we
consider distributions of digits of random variable $X$ with a given
distribution as well as distribution of random variable $\xi$
determined by distributions of digits of its $G_2$-representation
(\ref{ex:ab:1}) if the digits are independent.
Lebesgue structure of the probability distribution is described.
Differential properties of the probability distribution function as
well as properties of its spectrum and support are also studied.