On an application of Guth-Katz theorem

TL;DR

This paper applies the Guth-Katz theorem to establish lower bounds on the number of distinct triangle areas and dot products determined by point sets in the plane, leading to a sum-product estimate for real sets.

Contribution

It introduces a novel application of the Guth-Katz theorem to combinatorial geometry, deriving new bounds on geometric configurations and sum-product estimates.

Findings

At least cN/ log N distinct triangle areas with one vertex at the origin.

At least cN/ log N distinct dot products in the point set.

A sum-product bound |A·A ± A·A| ≥ c|A|^2 / log |A| for discrete sets A.

Abstract

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products. This in particular implies a sum-product bound $$ |A\cdot A\pm A\cdot A|\geq c\frac{|A|^2}{\log |A|} $$ for a discrete $A \subset {\mathbb R}$.