Incidences and pairs of dot products

Original · EN

Let F be a field, let P Fᵈ be a finite set of points, and let α,β∈ F {0}. We study the quantity |Πα, ᵦ| = {(p,q,r) ∈ P × P × P p · q = α, p · r = β}. We observe a connection between the question of placing an upper bound on |Πα,ᵦ| and a well-studied question on the number of incidences betwen points and hyperplanes, and use this connection to prove new and strengthened upper bounds on |Πα,ᵦ| in a variety of settings.