A generalization of the Jarník–Besicovitch theorem by continued fractions
A generalization of the Jarník–Besicovitch theorem by continued fractions
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DOI：
10.1017/etds.2014.98
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发表时间：
2015-02
影响因子：
0.9
通讯作者：
Bao-Wei Wang;Jun Wu;Jian Xu
中科院分区：
文献类型：
--
作者：
Bao-Wei Wang;Jun Wu;Jian Xu
我们应用连分数工具来解决丢番图近似，包括经典的贾尼克-贝西科维奇定理、局部贾尼克-贝西科维奇定理及其几种推广。众所周知，经典的 Jarník–Besicovitch 集合，用连分数表示，可以写成 $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{对于无穷多个}~n\in \mathbb{N}\},\end{eqnarray}$$ 其中 $T$ 是高斯映射，$a_{n}(x)$ 是 $x$ 的第 $n$ 部分商。在本文中，我们考虑广义 Jarník–Besicovitch 集合的大小 $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for 无穷多个}~n\in \mathbb{N}\},\end{eqnarray}$$ 其中 ${\it\tau}(x)$ 和 $f(x)$ 是在 $[0,1]$ 上定义的正函数。
We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where $T$ is the Gauss map and $a_{n}(x)$ is the $n$th partial quotient of $x$. In this paper, we consider the size of the generalized Jarník–Besicovitch set $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where ${\it\tau}(x)$ and $f(x)$ are positive functions defined on $[0,1]$.