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 Hölder 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, $$\sum _{n=1}^{\mathrm{\infty }}{\mathcal{L}}^d(R_n)=\mathrm{\infty }⟹\underset{n\to \mathrm{\infty }}{\lim }\frac{\sum _{k=1}^n{\chi }_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum _{k=1}^n{\mathcal{L}}^d(R_k)}=h(\mathbf{x})\text{for }\mu \text{-a.e.}\mathbf{\text{x}},$$ where $\mathbf{\text{x}}\in [0,1{]}^d$ and $R_k+\mathbf{\text{x}}$ is the translation of $R_k$. The result applies to Gauss map, $\beta$-transformation and expanding toral endomorphisms.