## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  222.10053
Autor:  Erdös, Paul; Guy, R.K.
Title:  Distinct distances between lattice points. (In English)
Source:  Elemente Math. 25, 121-123 (1970).
Review:  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 > n2/3-\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
Classif.:  * 11N56 Rate of growth of arithmetic functions

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