A generalization of the three-distance theorem for groups
type: Item
instance of: scholarly article
title: A generalization of the three-distance theorem for groups
zbMATH Open document ID: 0751.06009
DOI: 10.1007/BF01190760
author: Vera T. Sós
author: Ervin Fried
published in: Algebra Universalis
publication date: 1992-06-28T00:00:00Z
review text: The investigations of the present paper have been motivated by some results from the theory of diophantine approximations; cf. \textit{V. T. Sós} [Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 1, 127--134 (1958; Zbl 0094.02903)]. A group \(G\) with an order relation \(\leqq\) and a preorder relation \(\lesssim\) is called a group-lattice. The notions of interval, closed interval and of the lower neighbour \(L(v,I)\) of an element \(v\) in an interval \(I\) are defined by means of the relations \(\leqq\) and \(\lesssim\). A group-lattice \(G\) is called discrete (abbreviated: \(DG\)-lattice) if \(L(v,I)\) exists for every interval \(I\) and every \(v\in I\). If this condition is satisfied for every closed interval \(I\) and every \(v\in I\), then \(G\) is said to be closed discrete (abbreviated: \(CDG\)-lattice). The main results of the paper are ``the three distance theorems'' for \(DG\)-lattices and \(CDG\)-lattices; we quote the corresponding theorem for \(DG\)-lattices: Let \(I\) be an interval in the \(DG\)-lattice \(G\). Then the system \(\{x-L(x,I)\mid\;x\in I\}\) consists of at most three different values; and if it consists of exactly three different values, then one of them is the sum of the other two. Next, a structure theorem for archimedean \(CDG\)-lattices is deduced.
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 08A99
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 37910
zbMATH Keywords: discrete group-lattice
zbMATH Keywords: archimedean group-lattice
zbMATH Keywords: diophantine approximations
zbMATH Keywords: order
zbMATH Keywords: preorder
zbMATH Keywords: group-lattice
reviewed by: Ján Jakubík
MaRDI profile type: Publication
cites work: Q5509706
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cites work: On the theory of diophantine approximations. I1 (on a problem of A. Ostrowski)
cites work: On successive settings of an arc on the circumference of a circle
full work available at URL: https://doi.org/10.1007/bf01190760
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description: scientific article; zbMATH DE number 37910