论文标题
持续分数中连续vs单部分商的乘积对集合的Hausdorff维度分析
A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions
论文作者
论文摘要
我们提供了一组实数的详细的Hausdorff维度分析，其中连续的部分商在其持续的分数扩展中的乘积以一定的速度增长，但单个部分商的增长速度不同。我们考虑集合\ begin {equination*} \ ff（φ_1，φ_2）\ defeq \ ee（φ_1）\ backslash \ ee（φ_2）= \ left \ left \ weft \ {x \ in [0,1）： a_n（x）a_ {n+1}（x）＆\geqφ_1（n）\ text {\，\，对于无限的许多} n \ in \ n \ n \ n \ n \ n a_ {n+1}（x）＆<φ_2（n）\ text {\，\，对于所有足够大的} n \ in \ n \ end {split} \ right \}，\ end \ end {equation*}，其中$φ_i：$φ_i：\ n \ n \ to（\ n \ to） $ \ lim \ limits_ {n \ to \ infty}φ_i（n）= \ infty $。我们获得了一些令人惊讶的结果，包括$ \ ff（φ_1，φ_2）$的情况，对于$φ_i$'s的各种非平凡选择，都是空的。我们的结果通过推广几个已知结果，包括[非线性，33（6）：2615--2639，2020]的主要结果，从而有助于持续分数的度量理论。为了获得一些结果，我们考虑了一个替代的广义集，该集合可能具有独立感兴趣，并计算其Hausdorff维度。主要成分之一是使用经典的质量分布原理。特别是通过引入两种不同类型的概率度量的新想法，将质量仔细分布在cantor子集上。
We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions such that $\lim\limits_{n\to\infty} Φ_i(n)=\infty$. We obtain some surprising results including the situations when $\FF(Φ_1,Φ_2)$ is empty for various non-trivial choices of $Φ_i$'s. Our results contribute to the metrical theory of continued fractions by generalising several known results including the main result of [Nonlinearity, 33(6):2615--2639, 2020]. To obtain some of the results, we consider an alternate generalised set, which may be of independent interest, and calculate its Hausdorff dimension. One of the main ingredients is in the usage of the classical mass distribution principle; specifically a careful distribution of the mass on the Cantor subset by introducing a new idea of two different types of probability measures.
