Abstract: Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax$ (mod 1), where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\cdots\le|\lambda_d|$. We investigate the Hausdorff dimension of the recurrence set $$R(\psi):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,\psi(n)) for infinitely many n\}$$ for $\alpha\ge\log|\lambda_d/\lambda_1|$, where $\psi$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $\alpha=\liminf\limits_{n\to\infty}\frac{-\log \psi(n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.

Key words: quantitative recurrence properties, Hausdorff dimension, matrix transformations