On the Gaussian limiting distribution of lattice points in a parallelepiped

TL;DR

This paper proves that the normalized error term in counting lattice points within a parallelepiped, derived from a lattice in a totally real algebraic number field, converges to a Gaussian distribution as the size grows, with implications for low discrepancy sequences.

Contribution

It establishes the Gaussian limiting distribution for the lattice point error term in a parallelepiped for lattices from totally real algebraic number fields, extending understanding of distributional limits.

Findings

Normalized error term converges to Gaussian distribution as size increases.

Distributional limit applies to lattices from totally real algebraic number fields.

Results include implications for low discrepancy sequences.

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 $\Gamma$.