On the Gaussian limiting distribution of lattice points in a parallelepiped

Abstract

Let $ Γ\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)/σ(\cR,\bN) $ have Gaussian limiting distribution as $N\to \infty$, where $\btheta=(θ_1,...,θ_s)$ is a uniformly distributed random variable in $[0,1{]}^s$, $N=N_1...N_s$ and $σ(\cR,\bN) \asymp (\log N)^{(s-1)/2}$. We obtain also a similar result for the low discrepancy sequence corresponding to $\Gamma$.