A strong Borel–Cantelli lemma for recurrence

Consider a dynamical systems ([0, 1], T, µ) which is exponentially mixing for L1 against bounded variation. Given a non-summable sequence (mk) of non-negative numbers, one may define rk(x) such that µ(B(x, rk(x)) = mk. It is proved that for almost all x, the number of k ≤ n such that Tk(x) ∊ Bk(x) is approximately equal to m1+· · ·+mn. This is a sort of strong Borel–Cantelli lemma for recurrence. A consequence is that (Formula Presented) for almost every x, where τ is the return time.