On an application of Guth-Katz theorem

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 $A \subset {\mathbb R}$.