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arXiv · 1307.2076

On the Gaussian limiting distribution of lattice points in a parallelepiped

Mordechay B. Levin
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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 $[-θ_1 N_1,θ_1 N_1] \times ... \times [-θ_s N_s,θ_s N_{s}]$. 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 $Γ$.

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Mordechay B. Levin. 2013-07-08. On the Gaussian limiting distribution of lattice points in a parallelepiped. https://arxiv.org/abs/1307.2076

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