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 $$|A\cdot A\pm A\cdot A|\ge c\frac{|A{|}^2}{\log |A|}$$ for a discrete $A\subset \mathbb{R}$.

This paper has been withdrawn. A corrected version can be found at arXiv:1512.02670