Modified shrinking target problem for Matrix Transformations of Tori

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_{β_i}^n(x_i)-x_i| < ψ(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 $\mathrm{\mid }\cdot \mathrm{\mid }$ 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 ${\mathbb{T}}^d:={\mathbb{R}}^d\mathrm{/}{\mathbb{Z}}^d$. For any $1\le i\le d$, let ${\psi }_i$ be a positive function on $\mathbb{N}$ and $\mathrm{\Psi }(n):=({\psi }_1(n),\dots ,{\psi }_d(n))$ 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($\mathtt{x}$), Ψ(n)) $\text{ for infinitely many }$ n\in $\mathbb{N}$\big}, \end{equation*} where $L(f_n(\mathtt{x},\mathrm{\Psi }(n)))$ is a hyperrectangle and $\{f_n{\}}_{n\ge 1}$ is a sequence of Lipschitz vector-valued functions on ${\mathbb{T}}^d$ with a uniform Lipschitz constant.