Hausdorff dimension of recurrence sets for matrix transformations of tori

Let $T\text{\hspace{0.17em}⁣}:{\mathbb{T}}^d\to {\mathbb{T}}^d$, defined by $Tx=Ax(1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<\mathrm{\mid }{\lambda }_1\mathrm{\mid }\le \mathrm{\mid }{\lambda }_2\mathrm{\mid }\le \cdots \le \mathrm{\mid }{\lambda }_d\mathrm{\mid }$. We investigate the Hausdorff dimension of the recurrence set $$R(\psi ):=\{x\in {\mathbb{T}}^d\text{\hspace{0.17em}⁣}:T^{n}x\in B(x,\psi (n))\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{y}n\}$$ for $\alpha \ge \log \mathrm{\mid }{\lambda }_d\mathrm{/}{\lambda }_1\mathrm{\mid }$, where $\psi$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $\alpha =\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{-\log \psi (n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.