On the statistical properties of Klein polyhedra in three-dimensional lattices
type: Item
instance of: scholarly article
title: On the statistical properties of Klein polyhedra in three-dimensional lattices
zbMATH Open document ID: 1283.11097
author: A. A. Illarionov
published in: Sbornik: Mathematics
publication date: 2013-09-26T00:00:00Z
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H50
zbMATH DE Number: 6212107
zbMATH Keywords: lattice
zbMATH Keywords: Klein polyhedron
zbMATH Keywords: multidimensional continued fracitons
zbMATH Keywords: relative minimum
MaRDI profile type: Publication
cites work: Q5815662
cites work: Local minima of lattices and vertices of Klein polyhedra
cites work: Higher dimensional continued fractions
cites work: Q4821142
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cites work: Q4945497
cites work: On an invariant Möbius measure and the Gauss-Kuzmin face distribution
cites work: Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions
cites work: On the error of number-theoretic quadrature formulas
cites work: The discrepancy of the Korobov lattice points
cites work: The average number of vertices of Klein polyhedra for integer lattices
cites work: Average number of local minima for three-dimensional integral lattices
cites work: The average number of relative minima of three-dimensional integer lattices of a given determinant
cites work: On the asymptotic distribution of integer matrices
cites work: Klein polyhedra and relative minima of lattices
cites work: Estimates of the number of relative minima of lattices
full work available at URL: sm2013v204n06abeh004321
OpenAlex ID: W2061343696
DOI: 10.1070/SM2013V204N06ABEH004321
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review text: Let \(L\) be a complete three-dimensional lattice in \(\mathbb{R}^3\) and let \(O\) be one of the orthants in \(\mathbb{R}^3\). The boundary of the convex hull of all lattice points of \(L\) except for the origin is called a Klein polyhedron of the lattice \(L\). In fact, the Klein polyhedron is the set with the polyhedral structure. In this article the author obtains the formulae for the average value of the number of faces of a fixed type and of vertices of Klein polyhedra of three-dimensional lattice with a given determinant.
reviewed by: Oleg Karpenkov
description: scientific article; zbMATH DE number 6212107