AI 中文总结
本文研究有限字母集的无限连分数的值集的拓扑与度量性质，明确其各类集合属性与相关充要条件，证明了基本度量比率的有界性。
AI 中文摘要
本文研究所有形如 $0+\frac{1}{a_1+\dfrac{1}{a_2+_{\ddots}}}=1/a_1+1/a_2+...+1/a_n+...\equiv[0;a_1,a_2,...,a_n,...]$ 的无限连分数的值集 $E$ 的拓扑与度量性质，其部分商 $a_n$ 取值于有限正实数集 $\{e_0,e_1,...,e_{s-1}\}$，满足 $e_0<e_1<...<e_{s-1}$。研究确定了 $E$ 是有界集，基数为连续统，且是完备集；明确了 $E$ 为区间、疏朗集或勒贝格测度零集的条件；得到了 $E$ 为区间，且对应这类连分数编码系统具有零冗余（即每个数至多有两种表示）的充要条件；通过柱集（柱集与柱区间）的性质描述了该表示中数字的几何意义及度量关系；证明了由柱集直径与其父柱集直径之比定义的基本度量比率，既不趋近于零也不趋近于一。
英文摘要
The paper investigates the topological and metric properties of the set $E$ of values of all infinite continued fractions of the form \[0+\frac{1}{a_1+\dfrac{1}{a_2+_{\ddots}}}=1/a_1+1/a_2+...+1/a_n+...\equiv[0;a_1,a_2,...,a_n,...],\] whose partial quotients ($a_n$) take values in a finite set of positive real numbers $\{e_0,e_1,...,e_{s-1}\}$, $e_0<e_1<...<e_{s-1}$. It is established that the set $E$ is bounded, has cardinality continuum, and is perfect. Conditions under which $E$ is an interval, a nowhere dense set, or a set of Lebesgue measure zero are established. Necessary and sufficient conditions are obtained for $E$ to be an interval and for the corresponding coding system by such continued fractions to have zero redundancy, i.e., for each number to have at most two representations. The geometric meaning of the digits in this representation, as well as the metric relations, is described in terms of the properties of cylindrical sets (cylinders and cylindrical intervals). It is proved that the basic metric ratio, defined as the ratio of the diameter of a cylinder to the diameter of its parent cylinder, is bounded away from both zero and one.
