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 \[\left\{(u,v,w)\in E \times E \times E : u\cdot v = α, u \cdot w = β\right\}\] for units $\alpha$ and $\beta$.