论文标题
持续分数中多个部分商的加权产品的度量特性
Metrical properties for the weighted products of multiple partial quotients in continued fractions
论文作者
论文摘要
经典的khintchine和jarník定理是Dirichlet定理的结果的概括，是双苯胺近似理论的基本结果。这些定理与一组实际数字的大小有关，部分代理在其持续的分数扩展中以一定的速度增长。最近，观察到，在实际数字的持续扩展中，连续的部分商对的产物的增长与Dirichlet定理的改进有关。在本文中，我们考虑了几位连续偏向不同权力的局部局部产品的产品。也就是说，我们找到了以下集合的lebesgue度量和Hausdorff尺寸：$$ {\ d _ {\ MathBf t}}}}}}（ψ）：= \ left \ {x \ in [0，1）：\ prod \ prod \ limits_ \geψ（n）\ {\ text {对于无限的多个}}}}} \ n \ in \ n \ right \}，$ $ where $ t_i \ in \ mathbb r _+$ in \ mathbb r _+$ for all $ {0 \ leq i \ leq i \ leq m-1} $ {us
The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set: $$ {\D_{\mathbf t}}(ψ):=\left\{x\in[0, 1): \prod\limits_{i=0}^{m-1}{a^{t_i}_{n+i}(x)} \ge Ψ(n)\ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $t_i\in\mathbb R_+$ for all ${0\leq i\leq m-1}$, and $Ψ:\N\to\R_{\ge 1}$ is a positive function.
