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 ηlim infn→∞ (log Kn(x,y)/n) = 1 a.e., lim supn→∞ (log Kn(x,y)/n) = η a.e.Since the set of irrational numbers of type 1 has measure 1, for almost every θ the limit exists and is 1.