Discrepancy estimate of normal vectors

Let $A$ be an $s \times s$ invertible matrix with integer entries and with eigenvalues $|\lambda_i| > 1, i=1, \ldots,s$. In this paper we prove explicitly that there exists a vector $\alpha$, such that the discrepancy of the sequence $\{\alpha A^n\}_{n=1}^{N}$ is equal to $O(N^{-1} (\log N)^{2s+3})$ for $N \longrightarrow \infty$. This estimate can be improved no more than on the logarithmic factor.