  For any $\beta>1$, let $T_\beta$ be the classical $\beta$-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 $T_\beta^nx$ and $x_0$ is at
most equal to $\beta^{-N\hat{v}}$. This work extends the result of Bugeaud and
Liao \cite{YLiao2016} to every point $x_0$ in unit interval.
