The number of unit-area triangles in the plane: Theme and variations

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We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n²⁰/⁹), improving the earlier bound O(n⁹/⁴) of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be Ω(n²), for any triple of lines (it is always O(n²) in this case). (ii) We show that if S is a convex grid of the form A× B, where A, B are convex sets of n¹/² real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n³¹/¹⁴) unit-area triangles.