##
**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 > 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

**Classif.: ** * 11N56 Rate of growth of arithmetic functions

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag