Pairs of dot products in finite fields and rings

Abstract: We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a power of a prime. More precisely, given $E\subset \mathbb F_q^d$ or $\mathbb Z_q^d$, we provide bounds on the size of the set [\left{(u,v,w)\in E \times E \times E : u\cdot v = \alpha, u \cdot w = \beta \right}] for units $\alpha$ and $\beta$.