Circle rotations and the shrinking target properties

The shrinking target properties are related to recurrence. We will motivate
and present definitions of these properties.

We will also 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. We will give a detailed sketch of the proof.

Finally, we will apply our technique to give a new, very short, proof of the
logarithm law for irrational rotations.