On the refined shrinking target property of rotations

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

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