On Circle Rotations and the Shrinking Target Properties

We give a necessary and sufficient condition for a circle rotation to have the s-exponent monotone shrinking target property (sMSTP), and, thereby, we generalize a result for s = 1 that was established by J. Kurzweil and rediscovered by B. Fayad. As another application of our technique, we obtain a new, very short, proof of a known result, which concerns the behavior of irrational rotations and implies a logarithm law similar to D. Sullivan’s logarithm law for geodesics.