Gauss-Kuzmin problem for the difference Engel-series representation of real numbers (Q2680454)

Recall the classical Gauss-Kuszmin theorem: For \(x = [0;a_1,a_2,\ldots]\) written in its continued fraction, \(T([0;a_1,a_2,\ldots]) := T([0;a_2,a_3,\ldots])\) being the Gauss map, and \(W_n(a) = \{x \in (0,1]: \omega^n(x) < a\}\), it holds that for any \(a \in (0,1]\), \(\lim_{n \to \infty}\lambda(W_n(a)) = \log_2(1 + a)\). As the main result (Theorem 5) of this article, the author shows that for the difference representation of the Engel series, there is a degenerated behaviour of its underlying distribution in the following sense: For \(x \in [0,1)\), writing \[ x = \sum_{n=1}^{\infty} \frac{1}{(p_1(x)+1)\ldots(p_n(x)+1)}, \quad p_n \in \mathbb{N},\quad p_{n+1} \geq p_n \] in its classical Engel representation, let \begin{multline*} x = \Delta_{g_1(x)g_2(x)\ldots}^{\overline{E}} := \frac{1}{2+ g_1(x)} + \frac{1}{(2 + g_1(x))(2 + g_1(x) + g_2(x))} \\ +\frac{1}{(2 + g_1(x))(2 + g_1(x) + g_2(x))(2 + g_1(x) + g_2(x) + g_3(x))} + \ldots \end{multline*} with \(g_1(x) = p_1(x) -1, g_{n+1}(x) = p_{n+1}(x) - p_n(x), n \geq 1,\) denote the difference representation of the Engel series. Further, let \(\omega\) denote the left shift operator for this representation, that is, \(\omega(\Delta_{g_1(x)g_2(x)\ldots}^{\overline{E}}) := \Delta_{g_2(x)g_3(x)\ldots}^{\overline{E}}\) and let \(E_n(a) = \{x \in (0,1]: \omega^n(x) < a\}.\) Then for every \(a \in (0; 1]\), \[ \lim_{n \to \infty} \lambda(E_n(a)) = 1. \]