Канторвали як множини неелементарних ланцюгових дробів з обмеженим алфавітом

Abstract

Let $G_{\mathcal{A}}$ be a set of values of continued fractions whose elements belong to a bounded set $\mathcal{A}$ of positive real numbers. We prove that $G_{\mathcal{A}}$ is a continuum bounded and perfect set. For $\mathcal{A}_3=\{0{,}5; 1; 8\}$, the set $G_{\mathcal{A}}$ is a Cantorval, namely, it is homeomorphic to the set
$$E= \{x: x=\sum\limits_{k=1}^{\infty}(\frac{3\alpha_{2k-1}}{4^k}+\frac{2\alpha_{2k}}{4^k}),\alpha_k\in\{0,1\}\},$$
where $E$ contains a finite set of intervals whose complements are continuum nowhere dense sets.