论文标题
持续分数中的一些特殊的Borel-Bernstein定理
Some exceptional sets of Borel-Bernstein Theorem in continued fractions
论文作者
论文摘要
令$ [a_1（x），a_2（x），a_3（x），\ cdots] $表示[0,1）$中的实际数字$ x \的持续分数扩展。本文涉及到$ \ {a_n（x）\} _ {n \ geq1} $的增长率的某些特殊集。作为主要结果，集合\ [e _ {\ sup}（ψ）= \ left \ {x \ in [0,1）：\ \ limsup \ lims_ {n \ to \ frac} \ frac { 确定，其中$ψ：\ mathbb {n} \ rightArrow \ mathbb {r}^+$倾向于无限为$ n \ to \ infty $。
Let $[a_1(x),a_2(x), a_3(x),\cdots]$ denote the continued fraction expansion of a real number $x \in [0,1)$. This paper is concerned with certain exceptional sets of the Borel-Bernstein Theorem on the growth rate of $\{a_n(x)\}_{n\geq1}$. As a main result, the Hausdorff dimension of the set \[ E_{\sup}(ψ)=\left\{x\in[0,1):\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\} \] is determined, where $ψ:\mathbb{N}\rightarrow\mathbb{R}^+$ tends to infinity as $n\to\infty$.
