Lattices modulo \(N\) with long shortest distances
type: Item
instance of: scholarly article
title: Lattices modulo \(N\) with long shortest distances
zbMATH Open document ID: 1381.52025
DOI: 10.4171/EM/333
author: Florian Pausinger
published in: Elemente der Mathematik
publication date: 2017-11-17T00:00:00Z
review text: For coprime integers \(a\), \(b\) with \(0<a,b<N\) let \(\Pi_{N,a,b}\) be a set \(\{(na \pmod N, nb\pmod N):0\leq n < N \}\). \(\Pi_{N,a,b}\) can be considered as a lattice modulo \(N\). Denote by \(\lambda_1(\Pi_{N,a,b})\) the shortest distance between points in \(\Pi_{N,a,b}\). Further, suppose \(f_{\max}(N)=\max_{1\leq a,b < N}\) \(\lambda_1(\Pi_{N,a,b})/\sqrt{N}\). Then it is proved that for any \(\varepsilon>0\) there exist infinitely many \(N\) such that \(f_{\max}(N)>\sqrt{\frac{2}{\sqrt{3}}}-\varepsilon\). As a corollary, it is proved that for any \(\varepsilon>0\) there exists an integer matrix \(X\) such that \(\Delta(\Lambda_h)-\Delta(X\mathbb Z^2)<\varepsilon\), where \(\Lambda_h\) is a hexagonal lattice and \(\Delta(\cdot)\) is a lattice packing density.
Mathematics Subject Classification ID: 52C15
Mathematics Subject Classification ID: 52C05
zbMATH DE Number: 6808303
zbMATH Keywords: lattices
zbMATH Keywords: lattice approximations
zbMATH Keywords: shortest distances
reviewed by: Anton Shutov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/em/333
OpenAlex ID: W2737500553
description: scientific article; zbMATH DE number 6808303