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

We explore variants of ErdH 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 $1leq j leq k$.