  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.
