The spacing of the minima in certain cubic lattices
type: Item
instance of: scholarly article
title: The spacing of the minima in certain cubic lattices
zbMATH Open document ID: 0615.12003
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: From the author's paper: ''Let \({\mathcal K}\) be a cubic field with negative discriminant; let \(\mu,\nu\in {\mathcal K}\); and let \({\mathcal R}\) be a lattice with basis \(\{1,\mu,\nu\}\) such that 1 is a minimum of \({\mathcal R}\). If \(1=\theta_ 1\), \(\theta_ 2,\theta_ 3,...,\theta_ n,..\). is a chain of adjacent minima of \({\mathcal R}\) with \(\theta_{i+1}>\theta_ i\) \((i=1,2,3,...)\), then \(\theta_{n+5}\geq \theta_{n+3}+\theta_ n.\) This result can be used to prove that if p is the period of Voronoi's continued fraction algorithm for finding the fundamental unit \(\epsilon_ 0\) of \({\mathcal K}\), then \(\epsilon_ 0>\tau^{p/2}\), where \(\tau =(1+\sqrt{5})/2\). It is also shown that \(\theta_ n>4^{[(n- 1)/7]}.\)''
reviewed by: Claude Levesque
Mathematics Subject Classification ID: 12-04
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 3995817
zbMATH Keywords: cubic field
zbMATH Keywords: lattice
zbMATH Keywords: period
zbMATH Keywords: Voronoi's continued fraction algorithm
zbMATH Keywords: fundamental unit
author: Hugh C. Williams
MaRDI profile type: Publication
full work available at URL: pjm.1986.124.483
OpenAlex ID: W2061128863
DOI: 10.2140/PJM.1986.124.483
Recommended article: On cylindrical minima of three-dimensional lattices
Recommended article: The minimum distance parametrization of crystal lattices
Recommended article: Minimal densities of cubic sphere-packing types
Recommended article: Cylindrical minima of integral lattices
Recommended article: Q4884598
Recommended article: Packing minima and lattice points in convex bodies
Recommended article: Klein polyhedra and relative minima of lattices
Recommended article: Maximal spacings in several dimensions
Recommended article: Estimates of the number of relative minima of lattices
Recommended article: Minimal spacings of non-uniform densities
description: scientific article; zbMATH DE number 3995817