On the refined shrinking target property of rotations

We discuss the shrinking target property of irrational rotations. We obtain the condition of an irrational $\theta$ and monotone increasing $\varphi(n)$ such that $$ \liminf_{n \to \infty} n \varphi (n) \| n\theta - s \| = 0 \text{ for almost every } s. $$ We also consider the class of irrationals for which the limit inferior is 0 for every monotone increasing $\varphi(n)$ such that $\sum_n 1/(n\varphi(n))$ diverges.