A two-symbol system of encoding and some of its applications

Abstract

The article is devoted to a two-symbol system of encoding for real numbers with two bases of different signs \(g_0 \in (0;\frac{1}{2}]\) and \(g_1\equiv g_0-1\), as well as its applications in metric number theory and the metric theory of functions. We prove that any natural number \(a\) can be represented as $$a=2^n+\sum\limits_{k=1}^{n}[(-1)^{1+\sigma_k}a_k2^{n-k}]\equiv (1a_1\ldots a_n)_{G},$$ where \(a_k\in \{0;1\}\) and \(\sigma_k=a_1+\ldots+a_{k-1}\), and there exist exactly two such representations. Any number \(x\in (0;g_0]\) can be represented as

$$\delta_{\alpha_1}+\sum\limits_{k=2}^{\infty}(\delta_{\alpha_k}\prod\limits_{j=1}^{k-1}g_{\alpha_j})\equiv\Delta^{G_2}_{\alpha_1\alpha_2\ldots\alpha_k\ldots}, \delta_{\alpha_k}=\alpha_kg_{1-\alpha_k}.$$ Most numbers have a unique \(G_2\)-representation, while a countable set has exactly two representations: \(\Delta^{G_2}_{c_1\ldots c_m01(0)}=\Delta^{G_2}_{c_1\ldots c_m11(0)}\). For \(g_0=\frac12\), any number \(x\) in the interval \([0;1]\) has the expansion

$$x=\frac{1}{2}\alpha_0+\sum\limits_{k=1}^{\infty}\frac{\alpha_k(-1)^{1+\sigma_k}}{2^k}\equiv\Delta_{\alpha_0\alpha_1\ldots\alpha_n\ldots}.$$