Covering and packing in \({\mathbb Z^n}\) and \({\mathbb R^n}\). II
type: Item
instance of: scholarly article
title: Covering and packing in \({\mathbb Z^n}\) and \({\mathbb R^n}\). II
zbMATH Open document ID: 1247.11093
author: David M. Tuller
author: Wolfgang M. Schmidt
published in: Monatshefte für Mathematik
publication date: 2010-05-17T00:00:00Z
review text: In the present part (II), the authors deal with the group \( G=\mathbb Z^n\) and examine the effect of linear transformations on minimal covering and maximal packing densities of finite sets \(\mathcal A\subset \mathbb Z^n\). They show that the set of all densities for sets \(\mathcal A\) of given cardinality is closed and they caracterize four-element sets \(\mathcal A\subset \mathbb Z^n\) which are ''tiles'' see [\textit{W. M. Schmidt} and \textit{D. M. Tuller}, ``Covering and packings in \(\mathbb Z^n\) and \(\mathbb R^n\). I'', Monatsh. Math. 153, No. 3, 265--281 (2008; Zbl 1151.11029)] for the part (I). {Definitions:} Let \(\mathbb G\) be an additive group and \(\mathcal A\) a finite non empty subset. A further subset \(\mathcal S\) provides a \textit{covering} if the sum \(\mathcal A+\mathcal S=\mathbb G\). We say \(\mathcal S\) provides a \textit{packing} and write \(\mathcal A\downarrow \mathcal S\) if group elements can be written in at most one way as \(a+s\) with \(a\in\mathcal A, s\in\mathcal S\). We say that \(\mathcal S\) provides a \textit{tiling} if both \(\mathcal A+\mathcal S=\mathbb G\) and \(\mathcal A\downarrow \mathcal S\). A set \(\mathcal A\) is called a \textit{tile} if there exists a set \(\mathcal S\) with \(\mathcal A\oplus \mathcal S=\mathbb G\) Let \(\Lambda\subset \mathbb Z^n\) be a free \(\mathbb Z\)-module of finite, positive rank \(r\). A subset \(\mathcal S\subset \Lambda\) will be said \textit{periodic} with period \(q>0\) if \(\mathcal S\) is invariant under translation by elements \(q{\mathbf \ell} \) with \(\mathbf \ell\in\Lambda\). The \textit{density } \(d(S;\Lambda)\) of the set \(\mathcal S\) is the number of its images in the quotient \(\Lambda/q\Lambda\) divided by \(q^r\). Given \(\mathcal A\subset \Lambda\), the \textit{minimal covering frequency} \(c(\mathcal A;\Lambda)\) is the infimum of \(d(\mathcal S;\Lambda)\) over periodic sets with \(\mathcal A+\mathcal S=\Lambda\), and the \textit{maximal packing frequency} \(p…
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11H31
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 52C22
zbMATH DE Number: 5708708
zbMATH Keywords: lattice packings and coverings
zbMATH Keywords: tiling in n dimension
zbMATH Keywords: densities
MaRDI profile type: Publication
full work available at URL: s00605-009-0099-x
OpenAlex ID: W2064620507
cites work: Factoring polynomials with rational coefficients
cites work: Tesselation of integers
cites work: Lower bounds for heights on finitely generated groups
cites work: Covering and packing in \({\mathbb Z}^n\) and \({\mathbb R}^n\). I
DOI: 10.1007/S00605-009-0099-X
description: scientific article; zbMATH DE number 5708708