Inhomogeneous Diophantine approximation with general error functions

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

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