Recurrence speed of multiples of an irrational number

Abstract

Let $0 < \theta < 1$ be irrational and $T_{\theta} x = x + \theta \bmod 1$ on $[0,1)$. Consider the partition $\mathcal{Q}_n = \{[(i - 1) / 2^n, i/2^n) : 1 \leq i \leq 2^n\}$ and let $Q_n(x)$ denote the interval in $\mathcal{Q}_n$ containing $x$. Define two versions of the first return time: $J_n(x) = \min\{ j \geq 1 : \| x - {T_{\theta}}^j x \| = \| j \cdot \theta \| < 1/2^n \}$ where $\| t \| = \min_{n \in \mathbf{Z}} |t - n|$, and $K_n(x) = \min\{ j \geq 1 : {T_\theta}^j x \in Q_n(x) \}$. We show that $\log J_n / n \to 1$ and $\log K_n(x) / n \to 1$ a.e. as $n \to \infty$ for a.e. $\theta$.