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Distinct distances between lattice points. (English) Zbl 0222.10053

Let \(k\) be the greatest number of points in real \(2\)-space with integer coordinates between \(1\) and \(n\) and for which all mutual distances are distinct. By a simple counting argument, \(k \leq n\). For \(2 \leq n \leq 7\), \(k=n\) is verified by a choice of points in the plane. From a result of E. Landau [Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig, 1909), Bd. 2, p. 643] there is a positive constant \(c\) with \(k<cn(\log n)^{-1/4}\). A simple combinatorial proof is given that for \(\epsilon >0\), if \(n\) is sufficiently large, then \(k>n^{2/3-\epsilon}\). Results for dimensions \(1\) and \(3\) are mentioned.
Two problems are suggested: 1. Find the minimum number of points, determining distinct distances, so that no point may be added without duplicating a distance. 2. Given any \(n\) points in the plane (or \(d\)-space) how many can one select so that the distances are all distinct?
Reviewer: J.P.Tull

MSC:

11N56 Rate of growth of arithmetic functions
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