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.
Categories: math.DS