Upper bounds on pairs of dot products

Abstract

Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha, \beta)$, we bound the size of the set $$\left\{(p,q,r)\in P \times P \times P : p \cdot q = \alpha, p \cdot r = \beta \right\}.$$