Hausdorff dimension of sets with restricted, slowly growing partial quotients
Hausdorff dimension of sets with restricted, slowly growing partial quotients
复制标题
DOI：
10.1090/proc/15579
复制
发表时间：
2021-03
影响因子：
1
通讯作者：
Hiroki Takahasi
中科院分区：
文献类型：
--
作者：
Hiroki Takahasi
I.J.Good(1941)证明了偏商$a_n$趋于无穷的$(0，1)$中的无理数集具有Hausdorff维$1/2$。一些相关结果在B$或$a_n\geq f(N)$中施加了$a_n类型的限制，其中$B$是$\mathbb N$的无限子集，$f$是具有$n$的快速增长的函数。证明了对于取值于$[min B，inty)$且趋于无穷大的任意$B$和任意$f$，$(0，1)$中的无理数集使得：[a_n\in B，\a_n\leq f(N)\text{对于所有的$n\in\mathbb N$，和}a_n\to\infty\n\to\Infty\]是Hausdorff维$\tau(B)/2，$其中$\tau(B)$是$B$的收敛指数.
I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\geq f(n)$, where $B$ is an infinite subset of $\mathbb N$ and $f$ is a rapidly growing function with $n$. We show that, for an arbitrary $B$ and an arbitrary $f$ with values in $[\min B,\infty)$ and tending to infinity, the set of irrational numbers in $(0,1)$ such that \[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\] is of Hausdorff dimension $\tau(B)/2,$ where $\tau(B)$ is the exponent of convergence of $B$.