  We calculate the Hausdorff dimension of the fractal set \begin{equation*}
  \Big\{\mathtt{x}\in \mathbb{T}^d: \prod_{1\leq i\leq
d}|T_{\beta_i}^n(x_i)-x_i| < \psi(n) \text{ for infinitely many } n\in
\mathbb{N}\Big\}, \end{equation*} where the $T_{\beta_i}$ is the standard
$\beta_i$-transformation with $\beta_i>1$, $\psi$ is a positive function on
$\mathbb{N}$ and $|\cdot|$ is the usual metric on the torus $\mathbb{T}$.
  Moreover,
  we investigate a modified version of the shrinking target problem, which
unifies the shrinking target problems and quantitative
  recurrence properties for matrix transformations of tori. Let $T$ be a
$d\times d$ non-singular matrix with real coefficients. Then, $T$ determines a
self-map of the $d$-dimensional torus $\mathbb{T}^d:=\mathbb{R}^d /
\mathbb{Z}^d$.
  For any $1\leq i \leq d$, let $\psi_i$ be a positive function on $\mathbb{N}$
and $\Psi(n):=(\psi_1(n),\dots, \psi_d(n))$ with $n\in \mathbb{N}$. We obtain
the Hausdorff dimension of the fractal set
  \begin{equation*}
  \big\{\mathtt{x}\in \mathbb{T}^d: T^n(x)\in L(f_n(\mathtt{x}), \Psi(n))
\text{ for infinitely many } n\in \mathbb{N}\big\}, \end{equation*} where
$L(f_n(\mathtt{x}, \Psi(n)))$ is a hyperrectangle and $\{f_n\}_{n\geq 1}$ is a
sequence of Lipschitz vector-valued functions on $\mathbb{T}^d$ with a uniform
Lipschitz constant.
