Strong Borel--Cantelli Lemmas for Recurrence

Let $(X,T,\mu ,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there exist open balls $B_k(x)$ around $x$ such that $\mu (B_k(x))=M_k$. Under a short return time assumption, we prove a strong Borel–Cantelli lemma, including an error term, for recurrence, i.e., for $\mu$-a.e. $x\in X$, $$\sum _{k=1}^n{\mathbf{1}}_{B_k(x)}(T^{k}x)=\mathrm{\Phi }(n)+O(\mathrm{\Phi }(n{)}^{1\mathrm{/}2}(\log \mathrm{\Phi }(n){)}^{3\mathrm{/}2+\epsilon }),$$ where $\mathrm{\Phi }(n)={\sum }_{k=1}^n\mu (B_k(x))$. Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of ${\mathbf{T}}^2$.