On the Gaussian limiting distribution of lattice points in a parallelepiped

Abstract

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 $\Gamma$.