A generalization of the three-distance theorem for groups
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.
scientific article; zbMATH DE number 37910
scientific article