On the Minkowski distances and products of sum sets

Oliver Roche-Newton, Misha Rudnev

Research output: Contribution to journalArticle (Academic Journal)

11 Citations (Scopus)
252 Downloads (Pure)

Abstract

Given two points pq in the real plane, the signed area of the rectangle with the diagonal [pq] equals the square of the Minkowski distance between the points pq. We prove that N >1 points in the Minkowski plane ℝ1,1 generate Ω(NlogN)Ω(Nlog⁡N) 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 3D 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 3D 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.

As a corollary we establish the following near-optimal sum-product type estimate for finite sets AB ⊂ 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 languageUndefined/Unknown
Pages (from-to)507-526
Number of pages20
JournalIsrael Journal of Mathematics
Volume209
Issue number2
DOIs
Publication statusPublished - 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

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