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 [formula omitted] for some 1 ≤ i ≤ k, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.
Original language | English |
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Pages (from-to) | 401-408 |
Number of pages | 8 |
Journal | Combinatorics Probability and Computing |
Volume | 2 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1993 |
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ASJC Scopus subject areas
- Applied Mathematics
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
Cite this
Nearly Equal Distances in the Plane. / Erdős, P.; Makai, Endre; Pach, János.
In: Combinatorics Probability and Computing, Vol. 2, No. 4, 1993, p. 401-408.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Nearly Equal Distances in the Plane
AU - Erdős, P.
AU - Makai, Endre
AU - Pach, János
PY - 1993
Y1 - 1993
N2 - 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 [formula omitted] for some 1 ≤ i ≤ k, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.
AB - 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 [formula omitted] for some 1 ≤ i ≤ k, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.
UR - http://www.scopus.com/inward/record.url?scp=84971678134&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84971678134&partnerID=8YFLogxK
U2 - 10.1017/S0963548300000791
DO - 10.1017/S0963548300000791
M3 - Article
AN - SCOPUS:84971678134
VL - 2
SP - 401
EP - 408
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
SN - 0963-5483
IS - 4
ER -