HAUSDORFF DIMENSION FOR THE SET OF POINTS CONNECTED WITH THE GENERALIZED JARNÍK–BESICOVITCH SET
HAUSDORFF DIMENSION FOR THE SET OF POINTS CONNECTED WITH THE GENERALIZED JARNÍK–BESICOVITCH SET
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DOI：
10.1017/s1446788720000464
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发表时间：
2019-11
影响因子：
0.7
通讯作者：
Ayreena Bakhtawar
中科院分区：
文献类型：
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作者：
Ayreena Bakhtawar
本文研究了[0，1)$中点集$x的Hausdorff维数，使得对任意$r\in{N}$，$Begin{Align*}a_{n+1}(X)a_{n+2}(X)\cdots a_{n+r}(X)\geq e^{\tau(X)(h(X)+\cdots+h(T^{n-1}(X)}\end{align*}$对无穷多个$n\in{N}$成立，其中h和$\tau$是正连续函数，T是Gauss映射，$an}(X)$表示x在其连分式展开式中的第n次偏商.通过适当选择$r、tau(X)$和$h(X)$，我们得到了包括著名的Jarník-Besicovitch定理在内的各种经典结果。
In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$ , $$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$ holds for infinitely many $n\in \mathbb {N}$ , where h and $\tau $ are positive continuous functions, T is the Gauss map and $a_{n}(x)$ denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of $r,\tau (x)$ and $h(x)$ we obtain various classical results including the famous Jarník–Besicovitch theorem.