Gauss–Kuzmin problem for the difference Engel-series representation of real numbers

Abstract

UDC 511.7+517.5

Let $x=\Delta^{\overline{E}}_{g_1(x)\ldots g_n(x)\ldots}$ be the difference Engel-series representation of a real number $x\in\left(0;1\right]$ (${\overline{E}}$-representation), where $\Delta^{\overline{E}}_{g_1\ldots g_n\ldots}=\displaystyle\sum\nolimits_{n=1}^\infty\dfrac{1}{(2+g_1)\ldots(2+g_1+\ldots+g_n)},$ $\omega^n(x)=\Delta^{\overline{E}}_{g_{n+1}(x)g_{n+2}(x)\ldots}$ is an $n$-fold  operator of left shift  of digits in the $\overline{E}$-representation of the number $x$.  For a sequence of sets $E_n(a)=\left\{x\colon x\in\left(0;1\right),\omega^n(x)<a\right\}$, where $a$ is a fixed parameter with $\left(0;1\right]$, it is proved that $\lim_{n\to\infty} \lambda\left(E_n(a)\right)=1$, where $\lambda(\cdot)$ is a Lebesgue measure. This problem is similar to the classical Gauss–Kuzmin problem for elementary continued  fractions. However, their solutions  are noticeably different.