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

Abstract

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) $({\alpha }_1,\dots ,{\alpha }_k)$, 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 $1\le j\le k$.