### Abstract

Given two points *p*, *q* in the real plane, the signed area of the rectangle with the diagonal [*pq*] equals the square of the Minkowski distance between the points *p*, *q*. We prove that *N* >1 points in the Minkowski plane ℝ1,1 generate Ω(NlogN)Ω(NlogN) distinct distances, or all the distances are zero. The proof follows the lines of the Elekes/Sharir/Guth/Katz approach to the Erdős distance problem, analysing the 3*D* incidence problem, arising by considering the action of the Minkowski isometry group *ISO**(1, 1).

The signature of the metric creates an obstacle to applying the Guth/Katz incidence theorem to the 3*D* problem at hand, since one may encounter a high count of congruent line intervals, lying on null lines, or “light cones”, all these intervals having zero Minkowski length. In terms of the Guth/Katz theorem, its condition of the non-existence of “rich planes” generally gets violated. It turns out, however, that one can efficiently identify and discount incidences, corresponding to null intervals, and devise a counting strategy, where the rich planes condition happens to be just ample enough for the strategy to succeed.

*A*,

*B*⊂ ℝ, with more than one element:|(A±B)⋅(A±B)|≫|A||B|log|A|+log|B||(A±B)⋅(A±B)|≫|A||B|log|A|+log|B|.

Original language | Undefined/Unknown |
---|---|

Pages (from-to) | 507-526 |

Number of pages | 20 |

Journal | Israel Journal of Mathematics |

Volume | 209 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Sep 2015 |

### Bibliographical note

16pp. This is a new extended version of the paper. The previous one had a small gap in the part of the proof, dealing with rich planes, which has been corrected### Keywords

- math.CO
- math.NT
- 68R05, 11B75

## Cite this

*Israel Journal of Mathematics*,

*209*(2), 507-526. https://doi.org/10.1007/s11856-015-1227-z