On the refined shrinking target property of rotations

Abstract

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

13 pages