### Abstract

Let d_{1} ≤ d_{2} ≤ . . . ≤ d_{(n2)} denote the distances determined by n points in the plane. It is shown that min Σ_{i}(d_{i+1}-d_{i})^{2}=O(n ^{-6/7}), where the minimum is taken over all point sets with minimal distance d_{1} ≥ 1. This bound is asymptotically tight.

Original language | English (US) |
---|---|

Pages (from-to) | 111-124 |

Number of pages | 14 |

Journal | Combinatorica |

Volume | 19 |

Issue number | 1 |

State | Published - 1999 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*19*(1), 111-124.

**Uniformly distributed distances - A geometric application of Janson's inequality.** / Pach, János; Spencer, Joel.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 19, no. 1, pp. 111-124.

}

TY - JOUR

T1 - Uniformly distributed distances - A geometric application of Janson's inequality

AU - Pach, János

AU - Spencer, Joel

PY - 1999

Y1 - 1999

N2 - Let d1 ≤ d2 ≤ . . . ≤ d(n2) denote the distances determined by n points in the plane. It is shown that min Σi(di+1-di)2=O(n -6/7), where the minimum is taken over all point sets with minimal distance d1 ≥ 1. This bound is asymptotically tight.

AB - Let d1 ≤ d2 ≤ . . . ≤ d(n2) denote the distances determined by n points in the plane. It is shown that min Σi(di+1-di)2=O(n -6/7), where the minimum is taken over all point sets with minimal distance d1 ≥ 1. This bound is asymptotically tight.

UR - http://www.scopus.com/inward/record.url?scp=0040707342&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040707342&partnerID=8YFLogxK

M3 - Article

VL - 19

SP - 111

EP - 124

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -