The waiting time for irrational rotations

Let Tx = x + ?? (mod 1). Define Kn(x, y) = min{j ??? 1 : Tjy ???. Qn(x)}, where Qn(x) = [2 -ni,2-n(i + 1)) for 2-ni ??? x < 2 -n(i + 1). Then for irrational ?? of type ?? lim inf n??????log Kn(x, y)/n = 1 a.e., lim sup n??????log Kn(x, y)/n = ?? a.e. Since the set of irrational numbers of type 1 has measure 1, for almost every 9 the limit exists and is 1.

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