Dirichlet uniformly well-approximated numbers

Abstract

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\le n\le Q$, such that $\Vert n\theta -y\Vert <Q^{-\tau }$, where $\Vert \cdot \Vert$ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any $\tau >0$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of $\theta$. It is also proved that with respect to $\tau$, the only possible discontinuous point of the Hausdorff dimension is $\tau =1$.

35 pages