Let \(T\colon\mathbb{T}^d\to \mathbb{T}^d\) , defined by Tx = Ax (mod 1), where A is a d × d integer matrix with eigenvalues 1 < ∣λ1∣ ≤ ∣λ2∣ ≤ ⋯ ≤ ∣λd∣. We investigate the Hausdorff dimension of the recurrence set \(R(\psi):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,\psi(n)) {\rm ~for~infinitely~ many~}n\}\) for α ≥ log ∣λd/λ1∣, where ψ is a positive decreasing function defined on ℕ and its lower order at infinity is \(\alpha = \mathop {\lim \inf}\limits_{n \to \infty}{- \log \psi (n) \over n}\) . In the case that A is diagonalizable over ℚ with integral eigenvalues, we obtain the dimension formula.
