  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
$1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ 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
$\tau=1$.
