Recurrence speed of multiples of an irrational number

Abstract

Let 0 < <theta> < 1 be irrational and T(<theta>)x = x + theta mod 1 on (0, 1). Consider the partition Q(n) = {((i-1)/2(n), i/2(n)) : 1 less than or equal to i less than or equal to 2(n)} and let Q(n)(x) denote the interval in Q(n) containing x. Define two versions of the first return time: J(n)(x) = min{j greater than or equal to 1 : parallel tox - T(theta)(j)x parallel to = parallel toj . theta parallel to < 1/2(n)} where <parallel>t parallel to = min(n is an element ofZ) vertical bart -n vertical bar, and K-n(x) = min{j greater than or equal to 1 : T-theta(j) x is an element of Q(n)(x)}. We show that log J(n)/n --> 1 and log K-n(x)/n --> 1 a.e. as n --> infinity for a.e. theta