Three-point configurations determined by subsets of \({\mathbb{F}_q}^2\) via the Elekes-Sharir paradigm (Q2341926)

Let \(q\) be a prime power and \(\mathbb{F}_q\) denote the \(q\) element field. If \(q\equiv 3\) mod 4 and \(E\) is a subset of the affine plane over \(\mathbb{F}_q\) such that \(| E| >Cq^{7/4}\), then \(E\) determines a positive proportion of all congruence classes of triangles in the plane.