Modified shrinking target problem for Matrix Transformations of Tori

Abstract

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.