On the Minkowski distances and products of sum sets

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 ${\mathbb{R}}^{1,1}$ generate $\Omega (\frac{N}{\log N})$ distinct distances, or all the distances are zero. The proof follows the lines of the Elekes/Sharir/Guth/Katz approach to the Erd\H os distance problem, analysing the 3D incidence problem, arising by considering the action of the Minkowski isometry group $IS{O}^{\ast }(1,1)$. 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 $A,B\subset \mathbb{R}$, with more than one element: $$|(A\pm B)\cdot (A\pm B)|\gg \frac{|A||B|}{\log |A|+\log |B|}.$$

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

Cited by in corpus (2)

• A short proof of a near-optimal cardinality estimate for the product of a sum set
• A bound on the multiplicative energy of a sum set and extremal sum-product problems