The family of bi-Lipschitz classes of Delone sets in Euclidean space has the cardinality of the continuum
type: Item
instance of: scholarly article
title: The family of bi-Lipschitz classes of Delone sets in Euclidean space has the cardinality of the continuum
zbMATH Open document ID: 1301.51015
DOI: 10.1134/S0081543811080050
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-09-10T00:00:00Z
review text: For a metric space \(M\), we call a set \(A \subset M\) a Delone set if there exist positive integers \(R\) and \(r\) such that (1) \(d_M(x,y) \geq r\) for any \(x,y \in A\), and (2) \(\displaystyle{\cup_{x \in A}B_R(x)=M}\). If there is a constant \(\lambda \geq 1\) and a bijection \(F:A \rightarrow B\) for two Delone sets \(A \subset M_1\) and \(B \subset M_2\), satisfying \(\frac{1}{\lambda}d_{M_1}(x,y) \leq d_{M_2}(F(x),F(y)) \leq \lambda d_{M_1}(x,y)\) for all \(x,y \in A\), we say \(A\) and \(B\) are bi-Lipschitz equivalent. In this paper, the author proves that the family of bi-Lipschitz equivalence classes of Delone sets in \(\mathbb{E}^d(d \geq 2)\) has the cardinalty of the continuum.
reviewed by: Joonhyung Kim
Mathematics Subject Classification ID: 51K05
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 03E17
zbMATH DE Number: 6342469
zbMATH Keywords: Delone set
zbMATH Keywords: bi-Lipschitz class
author: Alexander Magazinov
MaRDI profile type: Publication
cites work: Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture
cites work: Homogeneous trees are bilipschitz equivalent
cites work: Separated nets in Euclidean space and Jacobians of biLipschitz maps
cites work: Lipschitz maps and nets in Euclidean space
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description: scientific article; zbMATH DE number 6342469