Quadratic estimates for the number of integer points in convex bodies
type: Item
instance of: scholarly article
title: Quadratic estimates for the number of integer points in convex bodies
zbMATH Open document ID: 1117.11055
DOI: 10.1007/BF02874638
author: Leonardo Colzani
author: Ilaria Rocco
author: Giancarlo Travaglini
publication date: 2007-02-02T00:00:00Z
review text: Let \(\Omega\) be a convex domain in \(\mathbb R^d\) with smooth boundary with positive Gaussian curvature and let \(\mathcal{D}(r,x)\) be the discrepancy between volume and number of points with integer coordinates in the dilated and translated domain \(r\Omega-x\). This discrepancy is a periodic function in \(x\) and so can be expanded into a multiple Fourier series on the torus \(\mathbb T^d=\mathbb R^d/\mathbb Z^d\). There are standard approximation techniques in fourier theory to handle such expansions which are only conditionally convergent. Using an approximation kernel and Poisson summation allows the estimation of \[ \left\{\int_{\mathbb T^d}|\mathcal{D}(r,x)|^2\,d\mu(x)\right\}^{1/2} \] where \(\mu(x)\) is a probability measure on the torus with Fourier coefficients \(\hat\mu(k)\) and leads to the upper bound \[ c\epsilon r^{d-1}+cr^{(d-1)/2}\left\{\sum_{k\in\mathbb Z^d} (1+\epsilon|k|)^{-\alpha}(1+|k|)^{-1}|\hat\mu(k)|\right\}^{1/2} \] where \(\alpha\) is a large positive number and \(\epsilon>0\) is chosen to minimise the sum. The estimate generalises classical results of Kendall for \(\int_{\mathbb T^d}|\mathcal{D}(r,x)|^2\,dx\) and Hlawka for \(\sup_x|\mathcal{D}(r,x)|\) and contains several other interesting special cases. A similar approach is also used to estimate the discrepancy with respect to dilations.
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 5121517
reviewed by: John H. Loxton
MaRDI profile type: Publication
cites work: On the lattice point problem for ellipsoids
cites work: Lattice point problems and distribution of values of quadratic forms
cites work: On the Number of Lattice Points in a Convex Set
cites work: Über Integrale auf konvexen Körpern. I, II
cites work: The mean lattice point discrepancy
cites work: Exponential Sums and Lattice Points III
cites work: On the Generalized "Birth-and-Death" Process
cites work: Mean square discrepancy bounds for the number of lattice points in large convex bodies
cites work: Mean lattice point discrepancy bounds. II: Convex domains in the plane
cites work: On the average order of the lattice rest of a convex body
cites work: Lattice points in large convex bodies
cites work: Lattice points in bodies with algebraic boundary
cites work: On the average order of the lattice rest of a convex planar domain
cites work: On the mean lattice point discrepancy of a convex disc
cites work: Q3142876
cites work: Q5523061
full work available at URL: https://doi.org/10.1007/bf02874638
OpenAlex ID: W2000292599
Recommended article: The number of integer points on Vinogradov's quadric
Recommended article: Counting integer points on quadrics with arithmetic weights
Recommended article: On the quermassintegrals of convex bodies
Recommended article: Mean square discrepancy bounds for the number of lattice points in large convex bodies
Recommended article: Q4011309
Recommended article: Integer points close to convex hypersurfaces
Recommended article: Upper and lower bound results on the convex hull of integer points in polyhedra
Recommended article: On integer points in polyhedra
Recommended article: Diophantine approximations and integer points of cones
Recommended article: Q4314749
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5121517