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Nearly Equal Distances in the Plane

Published online by Cambridge University Press:  12 September 2008

Paul Erdős ,
Endre Makai  and
János Pach
Paul Erdős
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences
Endre Makai
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences
János Pach
Affiliation:
Department of Computer Science, City College, New York, and Mathematical Institute of the Hungarian Academy of Sciences
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Abstract

For any positive integer k and ε > 0, there exist nk,ε, ck, e > 0 with the following property. Given any system of n > nk,ε points in the plane with minimal distance at least 1 and any t1, t2…, tk ≥ 1, the number of those pairs of points whose distance is between ti and for some 1 ≤ i ≤ k, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 401 - 408
Copyright
Copyright © Cambridge University Press 1993

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References

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