A strong Borel--Cantelli lemma for recurrence

Abstract: 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.