### Abstract

For any positive integer k and ε > 0, there exist n^{k,ε}, c^{k, e} > 0 with the following property. Given any system of n > n_{k,ε} points in the plane with minimal distance at least 1 and any t_{1}, t_{2}…, t_{k} ≥ 1, the number of those pairs of points whose distance is between t_{i} and [formula omitted] for some 1 ≤ i ≤ k, is at most (n^{2}/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.

Original language | English |
---|---|

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

*Combinatorics Probability and Computing*,

*2*(4), 401-408. https://doi.org/10.1017/S0963548300000791

**Nearly Equal Distances in the Plane.** / Erdős, P.; Makai, Endre; Pach, János.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 2, no. 4, pp. 401-408. https://doi.org/10.1017/S0963548300000791

}

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 -