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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (\frac{N}{{\log N}})$$\end{document} 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).