Quantitative recurrence properties for piecewise expanding maps on $ [0,1]^d $

Let $T\colon[0,1]^d\to [0,1]^d$ be a piecewise expanding map with an absolutely continuous invariant measure $\mu$. Let $\{H_n\}$ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $\mathcal R(\{H_n\})$ the set of points $\mathbf x$ such that $T^n\mathbf x\in \mathbf x+H_n$ for infinitely many $n\in\mathbb N$, where $\mathbf x+H_n$ is the translation of $H_n$. We prove that if $\mu$ is exponential mixing and the density of $\mu$ is sufficiently regular, then the $\mu$-measure of $\mathcal R(\{H_n\})$ is zero or full according to the sum of the volumes of $H_n$ converges or not. In the case that $T$ is a matrix transformation, our results extend a previous work of Kirsebom, Kunde, and Persson [to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., 2023] in two aspects: by allowing the matrix to be non-integer and by allowing the `target' sets $H_n$ to be hyperrectangles or hyperboloids. We also obtain a dimension result when $T$ is a diagonal matrix transformation.