On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Mordechay B. Levin

doi:10.1515/udt-2016-0014

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On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Uniform distribution theory

Volume 11 (2016): Issue 2 (December 2016)

By: Mordechay B. Levin

Open Access

|Jan 2017

Abstract

Let Γ ⊂ ℝ^s 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] × . . . × [−θ_s N_s, θ_s N_s]. In this paper, we prove that ℛ(θ, N)/σ(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ₁, . . . , θ_s) is a uniformly distributed random variable in [0, 1]^s, N = N₁ . . . . N_s 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.

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Articles in this issue

DOI: https://doi.org/10.1515/udt-2016-0014 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 45 - 89

Submitted on: Sep 10, 2015

Accepted on: Oct 19, 2015

Published on: Jan 13, 2017

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

lattice points problem,

low discrepancy sequences,

totally real algebraic number field,

central limit theorem

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2017 Mordechay B. Levin, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 11 (2016): Issue 2 (December 2016)