标题： 连分数的度量理论带有多个大的部分商 显示英文标题

标题： Metric Theory for Continued Fractions with Multiple Large Partial Quotients

摘要： 在连分数的度量理论中，大分母项的存在可能会使许多经典极限定理失效。 一种常用的解决方法是在制定这些定理的变体形式时舍弃最大的分母项。 然而，当处理至少两个大的分母项时，这种方法会失效。 受Tan、Tian和Wang[Sci. China Math., 2023]最近工作的启发，我们研究了在连分数展开的前$n$项中至少包含$r$个大分母项的实数的度量理论。 具体来说，设$[a_1(x), a_2(x), \ldots ]$是实数$x \in [0, 1)$的连分数展开。 我们确定了以下集合的勒贝格测度和豪斯多夫维数：\[ F(r, \psi)=\Big\{ x \in [0,1): \exists 1 \leq k_1< \cdots < k_r \leq n, a_{k_i} (x) \geq \psi(n)~(i=1, \ldots, r)\ \text{for i.m.}~n \in \mathbb{N} \Big\}, \]其中 `i.m.' 表示“无限多个”，$r \geq 1$和$\psi$是定义在$\mathbb{N}$上的正函数。 显示英文摘要

摘要： The presence of large partial quotients can invalidate many classical limit theorems in the metric theory of continued fractions. A commonly employed strategy to overcome this problem is to discard the largest partial quotient when formulating variant forms of such theorems. However, this method will fail when dealing with at least two large partial quotients. Motivated by recent work of Tan, Tian, and Wang [Sci. China Math., 2023], we investigate the metric theory of real numbers that contain at least $r$ large partial quotients among the first $n$ terms of their continued fraction expansions. Specifically, let $[a_1(x), a_2(x), \ldots ]$ be the continued fraction expansion of a real number $x \in [0, 1)$. We determine the Lebesgue measure and Hausdorff dimension of the following set: \[ F(r, \psi)=\Big\{ x \in [0,1): \exists 1 \leq k_1< \cdots < k_r \leq n, a_{k_i} (x) \geq \psi(n)~(i=1, \ldots, r)\ \text{for i.m.}~n \in \mathbb{N} \Big\}, \] where `i.m.' stands for `infinitely many', $r \geq 1$ and $\psi$ is a positive function defined on $\mathbb{N}$.