论文标题
与β-转化有关
Uniform Diophantine approximation related to beta-transformations
论文作者
论文摘要
对于任何$β> 1 $，令$t_β$是经典的$β$转换。修复[0,1] $中的$ x_0 \和一个非负实数$ \ hat {v} $，我们计算了一组实际数字$ x \ in [0,1] $的hausdorff尺寸，该属性与$ n $相处的每个$ n $ n $ n $ n $ n $ n $，n $ n $ n $ \ leq n $ cyq n $， $ x_0 $最多等于$β^{ - n \ hat {v}} $。这项工作将Bugeaud和liao \ cite {yliao2016}的结果扩展到每个点$ x_0 $的单位间隔。
For any $β>1$, let $T_β$ be the classical $β$-transformations. Fix $x_0\in[0,1]$ and a nonnegative real number $\hat{v}$, we compute the Hausdorff dimension of the set of real numbers $x\in[0,1]$ with the property that, for every sufficiently large integer $N$, there is an integer $n$ with $1\leq n\leq N$ such that the distance between $T_β^nx$ and $x_0$ is at most equal to $β^{-N\hat{v}}$. This work extends the result of Bugeaud and Liao \cite{YLiao2016} to every point $x_0$ in unit interval.
