Pairs of dot products in finite fields and rings

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 $α$ and $β$.

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