Let $\tau$ be an ergodic measure-preserving map of a probability space $(\Omega, \mu)$.

• (Returns at positive density.) If $\mu(G) > 0$ and $\eta > 0$, then for almost every $\omega$ and all large $n$ there is $k \in [n, (1+\eta)n]$ with $\tau^k\omega \in G$.
• (The density argument.) Let $a_n : \Omega \to \mathbb{R}$ be measurable, $\varphi \ge 0$ measurable and integrable, and $c \in \mathbb{R}$. Suppose that almost surely, along the orbit of $\omega$,$$a_k(\tau^k\omega) \le a_n(\tau^n\omega)\, \exp \sum_{n < t \le k} \varphi(\tau^t\omega) \qquad (n \le k),$$and that for almost every $\zeta$ and every $b < c$, $a_n(\zeta) \ge e^{bn}$ for all large $n$. Then for almost every $\omega$ and every $b < c$, $a_n(\tau^n\omega) \ge e^{bn}$ for all large $n$.

Applied with $\tau = \sigma^{-1}$ to the growth of a cocycle, the second statement turns forward lower bounds into backward ones without any integrability of the inverse. It is the argument of Mañé's lemma on tempered functions (Mañé 1983; González-Tokman and Quas 2015, Lemma 15).