Discrepancy estimate of normal vectors (the case of hyperbolic matrices)

Let $A$ be a $t \times t$ invertible matrix with integer entries and with eigenvalues $|\lambda_i| \neq 1, i \in [1, t]$. In this paper we prove explicitly that there exists a vector $\alpha$, such that discrepancy of the sequence $\{\alpha A^n\}_{n=1}^{N}$ is equal to $O(N^{-1} (\log N)^{t+5})$ for $N \longrightarrow \infty$. This estimate can be improved no more than on the logarithmic factor.