Dirichlet一致好逼近数
Dirichlet uniformly well-approximated numbers
摘要：
修正无理数$θ$。对于一个实数$τ>0$，考虑数$y$满足对于所有的大数$Q$，存在一个整数$1\leq n\leq Q$，使得$\|nθ-y\|<Q^{-τ}$，其中$\|\cdot\|$是实数到其最近整数的距离。这些数被称为狄利克雷一致好逼近数。对于任何$τ>0$，Haussdorff维数的这些数字的集合获得，并示出依赖于丢番图性质的$θ$。证明了关于$τ$，豪斯多夫维数唯一可能的间断点是$τ=1$ .
英文摘要：
Fix an irrational number $θ$. For a real number $τ>0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|nθ-y\|<Q^{-τ}$, where $\|\cdot\|$ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any $τ>0$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of $θ$. It is also proved that with respect to $τ$, the only possible discontinuous point of the Hausdorff dimension is $τ=1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
