The waiting time for irrational rotations

Let Tx = x + θ (mod 1). Define K_n(x, y) = min{j ≥ 1 : T^jy ∈. Q_n(x)}, where Q_n(x) = [2 ^-ni,2^-n(i + 1)) for 2^-ni ≤ x < 2 ^-n(i + 1). Then for irrational θ of type η lim inf _n→∞log K_n(x, y)/n = 1 a.e., lim sup _n→∞log K_n(x, y)/n = η a.e. Since the set of irrational numbers of type 1 has measure 1, for almost every 9 the limit exists and is 1.