<abstract xmlns="http://www.w3.org/1999/xhtml">Let Γ ⊂ ℝ<italic>s</italic> be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(<bold><italic>θ</italic></bold>, <bold>N</bold>) be the error term in the lattice point problem for the parallelepiped [−<italic>θ</italic>1<italic>N</italic>1, <italic>θ</italic>1<italic>N</italic>1] × . . . × [−<italic>θs Ns</italic>, <italic>θs Ns</italic>]. In this paper, we prove that ℛ(<bold><italic>θ</italic></bold>, <bold>N</bold>)<italic>/σ</italic>(ℛ, <bold>N</bold>) has a Gaussian limiting distribution as <italic>N</italic>→∞, where <bold><italic>θ</italic></bold> = (<italic>θ</italic>1, . . . , <italic>θs</italic>) is a uniformly distributed random variable in [0, 1]<italic>s</italic>, <italic>N</italic> = <italic>N</italic>1 . . . . <italic>Ns</italic> and <italic>σ</italic>(ℛ, <bold>N</bold>) ≍ (log <italic>N</italic>)(<italic>s</italic>−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the <italic>S</italic>-unit theorem.</abstract>

Let Γ ⊂ ℝs be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the parallelepiped [−θ1N1, θ1N1] × . . . × [−θs Ns, θs Ns]. In this paper, we prove that ℛ(θ, N)/σ(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ1, . . . , θs) is a uniformly distributed random variable in [0, 1]s, N = N1 . . . . Ns and σ(ℛ, N) ≍ (log N)(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900,

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped"}

Let Γ ⊂ ℝs be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the...

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Publié en ligne: 13 janv. 2017

Pages: 45 - 89

Reçu: 10 sept. 2015

Accepté: 19 oct. 2015

DOI: https://doi.org/10.1515/udt-2016-0014

Mots clés
lattice points problem, low discrepancy sequences, totally real algebraic number field, central limit theorem

© 2016 Mordechay B. Levin, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Publié en ligne: 13 janv. 2017

Pages: 45 - 89

Reçu: 10 sept. 2015

Accepté: 19 oct. 2015

DOI: https://doi.org/10.1515/udt-2016-0014

Mots cléslattice points problem, low discrepancy sequences, totally real algebraic number field, central limit theorem

© 2016 Mordechay B. Levin, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
lattice points problem, low discrepancy sequences, totally real algebraic number field, central limit theorem