  Consider a mixing dynamical systems $([0,1], T, \mu)$, for instance a
piecewise expanding interval map with a Gibbs measure $\mu$. Given a
non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$
such that $\mu (B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the
number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to
$m_1 + \ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for
recurrence.
  A consequence is that \[
  \lim_{r \to 0} \frac{\log \tau_{B(x,r)} (x)}{- \log \mu (B (x,r))}
  = 1 \] for almost every $x$, where $\tau$ is the return time.
