Quantitative recurrence properties and strong dynamical Borel-Cantelli lemma for dynamical systems with exponential decay of correlations

Let $([0,1]^d,T,\mu)$ be a measure-preserving dynamical system so that the correlations decay exponentially for H\"older continuous functions. Suppose that $\mu$ is absolutely continuous with a density function $h\in L^q(\mathcal L^d)$ for some $q>1$, where $\mathcal L^d$ is the $d$-dimensional Lebesgue measure. Under mild conditions on the underlying dynamical system, we obtain a strong dynamical Borel-Cantelli lemma for recurrence: For any sequence $\{R_n\}$ of hyperrectangles with sides parallel to the axes and centered at the origin, $$\sum_{n=1}^{\infty}\mathcal L^d(R_n)=\infty\quad\Longrightarrow\quad\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\chi_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum_{k=1}^{n}\mathcal L^d(R_k)}=h(\mathbf{x})\quad\text{for $ \mu $-a.e.$\textbf{x}$},$$ where $\textbf{x}\in[0,1]^d$ and $R_k+\textbf{x}$ is the translation of $R_k$. The result applies to Gauss map, $\beta$-transformation and expanding toral endomorphisms.