Areas of rectangles and product sets of sum sets

Details Areas of rectangles and product sets of sum sets Areas of rectangles and product sets of sum sets

2012-03-28

arxiv.org/abs/1203.6237v1

Mathematics

Combinatorics

13pp

Scientific paper

Given two points $p,q\in{\mathbb{R}^2}$, the area of the rectangle with points $p$ and $q$ at opposite corners can be calculated. In this paper, it is proven that $N$ points in the plane generate $\Omega(\frac{N}{\log{N}})$ rectangles with distinct areas, or alternatively all areas are zero. Consequently, the following near-optimal sum-product type estimate for discrete sets of reals $A,B$ is obtained: $$|(A\pm{B})\cdot{(A\pm{B})}|\gg{\frac{|A||B|}{\log{|A|}+\log{|B|}}}.$$ The signed area of the rectangle with the diagonal $[pq]$ equals the square of the Minkowski distance between the points $p,q,$ and the proof of our main result is in essence an adaptation of the Guth-Katz proof of the Erd\H os distance conjecture to the case of the Minkowski distance. It requires a certain amount of care, however, because one of the assumptions of the Guth-Katz incidence theorem in $\R^3$ may get violated violated, owing to the possibility of having many zero Minkowski distances.

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