MODIFIED SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI

In this paper, we calculate the Hausdorff dimension of the fractal set $$\left\{x\in {𝕋}^d:\prod _{1\le i\le d}|T_{{\beta }_i}^n(x_i)-x_i|<\psi (n)\text{for infinitely many }n\in \mathbb{N}\right\},$$ where $T_{{\beta }_i}$ 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 $𝕋$. 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 ${𝕋}^d:={ℝ}^d/{\mathbb{Z}}^d$. 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 $$\{x\in {𝕋}^d:T^n(x)\in L(f_n(x),\mathrm{\Psi }(n))\text{for infinitely many }n\in \mathbb{N}\},$$ where $L(f_n(x,\mathrm{\Psi }(n)))$ is a hyperrectangle and ${\{f_n\}}_{n\ge 1}$ is a sequence of Lipschitz vector-valued functions on ${𝕋}^d$ with a uniform Lipschitz constant.