Inhomogeneous Diophantine approximation with general error functions

Abstract

Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ 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_{\varphi}(\al):={y\in \R: |n\al -y| < \varphi(n) \text{for infinitely many} n},] where $|\cdot|$ denotes the distance to the nearest integer.