On the number of unit-area triangles spanned by convex grids in the plane

Abstract

A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A,B\subset \mathbb{R}$, each of size $n^{1/2}$, the convex grid $A\times B$ spans at most $O(n^{37/17}{\log }^{2/17}n)$ unit-area triangles. This improves the best known upper bound $O(n^{31/14})$ recently obtained in \cite{RS}. Our analysis also applies to more general families of sets $A$, $B$, known as sets of Szemerédi--Trotter type.

arXiv admin note: substantial text overlap with arXiv:1501.00379

References in corpus (2)

• On sums of Szemerédi--Trotter sets
• The number of unit-area triangles in the plane: Theme and variations