The waiting time for irrational rotations

Abstract

Let Tx = x + θ (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 θ of type η lim inf n→∞log Kn(x, y)/n = 1 a.e., lim sup n→∞log Kn(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.