The waiting time for irrational rotations

摘要

Let Tx = x + theta (mod 1). Define K-n (x, y) = min {j greater than or equal to 1 : T(j)y is an element of Qn (x)}, where Q(n)(x) = [2(-n)i, 2(-n) (i + 1)) for 2(-n)i less than or equal to x < 2(-n)(i + 1). Then for irrational theta of type eta lim(n-->infinity) inf log K-n (x, y) / n = 1 a.e., lim(n-->infinity) sup log K-n (x, y) / n = eta a.e. Since the set of irrational numbers of type 1 has measure 1, for almost every theta the limit exists and is 1. [References: 17]