We generalize the monotone shrinking target property (MSTP) to the s-exponent
monotone shrinking target property (sMSTP) and give a necessary and sufficient
condition for a circle rotation to have sMSTP.
Using another variant of MSTP, 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.Comment: 13 pages. A new section has been added. The rest of the paper remains
the same except for some very minor revisions
