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

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⊂R, each of size n1/2, the convex grid A×B spans at most O(n37/17log2/17⁡n) unit-area triangles. Our analysis also applies to more general families of sets A, B, known as sets of Szemerédi-Trotter type.