Abstract
We generalize the monotone shrinking target property (MSTP) to the $s$-exponent monotone shrinking target property ($s$MSTP) and give a necessary and sufficient condition for a circle rotation to have $s$MSTP.   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.
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