On the number of dot product chains in finite fields and rings

We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime. Specifically, given a large finite set of points $E$, and a sequence of elements of the base field (or ring) $(\alpha_1,\ldots,\alpha_k)$, we give conditions guaranteeing the expected number of $(k+1)$-tuples of distinct points $(x_1,\dots, x_{k+1})\in E^{k+1}$ satisfying $x_j \cdot x_{j+1}=\alpha_j$ for every $1\leq j \leq k$.