On the Minkowski distances and products of sum sets

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(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 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|.