  Let $ \Gamma \subset \RR^s $ be a lattice obtained from a module in a totally
real algebraic number field. Let $\cR(\btheta, \bN)$ be an error term in the
lattice point problem for the parallelepiped $[-\theta_1 N_1,\theta_1 N_1]
\times ... \times [-\theta_s N_s,\theta_s N_{s}]$. In this paper, we prove that
$\cR(\btheta, \bN)/\sigma(\cR,\bN) $ have Gaussian limiting distribution as $N
\to \infty$, where $\btheta=(\theta_1,...,\theta_s)$ is a uniformly distributed
random variable in $[0,1]^s$, $N=N_1 ... N_s$ and $\sigma(\cR,\bN) \asymp (\log
N)^{(s-1)/2}$. We obtain also a similar result for the low discrepancy sequence
corresponding to $\Gamma$.
