Incidences and pairs of dot products

Abstract: Let $\mathbb{F}$ be a field, let $P \subseteq \mathbb{F}^d$ be a finite set of points, and let $\alpha,\beta \in \mathbb{F} \setminus {0}$. We study the quantity [|\Pi_{\alpha, \beta}| = {(p,q,r) \in P \times P \times P \mid p \cdot q = \alpha, p \cdot r = \beta }.] We observe a connection between the question of placing an upper bound on $|\Pi_{\alpha,\beta}|$ 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 $|\Pi_{\alpha,\beta}|$ in a variety of settings.