A discrete form of the Beckman-Quarles theorem for rational spaces
type: Item
instance of: scholarly article
title: A discrete form of the Beckman-Quarles theorem for rational spaces
zbMATH Open document ID: 1006.51012
author: Joseph Zaks
published in: Journal of Geometry
publication date: 2003-02-18T00:00:00Z
review text: Let \(A=\{4k(k+1):k=1,2,3,\dots \} \cup (\{k^2: k=1,2,3,\dots \} \cap \{2k^2-1: k=2,3,4,\dots \})\). The author proves: if \(n \in A\) and \(x,y \in \mathbb Q^n\), then there exists a finite set \(\{x,y\} \subseteq S_{xy} \subseteq \mathbb Q^n\) such that for every unit-distance preserving mapping \(f:S_{xy} \to \mathbb Q^n\), \(\|f(x)-f(y)\|=\|x-y\|\) holds. The theorem for \(n=8\) was earlier proved by \textit{A. Tyszka} [Aequationes Math. 62, No. 1-2, 85-93 (2001; Zbl 1004.51023)].
reviewed by: Apoloniusz Tyszka
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 05C12
zbMATH DE Number: 1760637
zbMATH Keywords: Beckman-Quarles theorem
zbMATH Keywords: isometry
zbMATH Keywords: unit-distance preserving mapping
zbMATH Keywords: unit-distance graph
MaRDI profile type: Publication
full work available at URL: s00022-001-8581-1
OpenAlex ID: W2061549458
DOI: 10.1007/S00022-001-8581-1
description: scientific article; zbMATH DE number 1760637