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. $\theta$.