Inhomogeneous Diophantine approximation with general error functions

Let $\text{al}$ be an irrational and $\phi :\mathbb{N}\to {\mathbb{R}}^{+}$ be a function decreasing to zero. For any $\text{al}$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_{\varphi}(\al):={y\in \R: |n\al -y| < \varphi(n) $\text{for infinitely many}$ n},] where $\mathrm{\mid }\cdot \mathrm{\mid }$ denotes the distance to the nearest integer.