Denote by $\|\cdot\|$ the distance of a real number to its nearest integer. Fix an irrational $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q\gg1$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\| < Q^{-\tau}$. 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 irrationality exponent $w(\theta):=\sup\{s > 0: \liminf_{j\to\infty}j^{s}\|j\theta\|=0\}$ of $\theta$. It is also proved that with respect to $\tau$, the only possible discontinuous point of the Hausdorff dimension is $\tau=1$.