Upper Bounds on Pairs of Dot Products

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 nonzero real numbers, \( (\alpha, \beta) \), we bound the size of the set \[ \{(p, q, r) \in P \times P \times P : p \cdot q = \alpha, p \cdot r = \beta\}. \]