On an Application of Guth–Katz Theorem

Abstract

We prove that for some universal c, a non-collinear set of N > 1 c points in the Euclidean plane determines at least c N log N distinct areas of triangles with one vertex at the origin, as well as at least c N log N distinct dot products.This in particular implies a sum-product boundsuch that (i) no more than O(N ) lines are concurrent, (ii) no more than O(N ) lines lie in a single plane, (iii) no more than O(N ) lines lie in a single doubly ruled surface, have O N 3 log N pair-wise intersections.Remark 1.1.It is well known that there are only two doubly ruled non-plane surfaces in R 3 : hyperbolic paraboloid, and single-sheeted hyperboloid, both having degree two (see, e.g., [14]).In this paper, we shall see that Theorem 1.1 implies the following results.Theorem 1.2.There exists a universal c > 0 such that a set of N > 1 c non-collinear points in R 2 determines at least c N log N distinct areas of triangles with one vertex at the origin.

Locations

• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - View - PDF