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

TL;DR

This paper establishes a new upper bound on the number of unit-area triangles formed by convex grids in the plane, improving previous bounds and extending to sets of Szemerédi–Trotter type.

Contribution

It provides the first non-trivial upper bound for convex grids, advancing understanding of geometric configurations in combinatorial geometry.

Findings

New upper bound of O(n^{37/17} log^{2/17} n) for unit-area triangles

Applicable to sets of Szemerédi–Trotter type

Improves previous bound of O(n^{31/14})

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 Szemer\'edi--Trotter type.