On the number of unit-area triangles spanned by convex grids in the plane
Raz, Orit E. · Sharir, Micha · Shkredov, Ilya D.
Original · EN
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, each of size n¹/², the convex grid A× B spans at most O(n³⁷/¹⁷²/¹⁷n) unit-area triangles. This improves the best known upper bound O(n³¹/¹⁴) recently obtained in RS. Our analysis also applies to more general families of sets A, B, known as sets of Szemerédi--Trotter type.
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