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

TL;DR

This paper investigates the number of chains formed by dot product relations in finite fields and rings, extending Erdős' unit distance problem to algebraic structures like _q^d and Z_q^d, and provides conditions for expected chain counts.

Contribution

It introduces new conditions ensuring the expected number of dot product chains in large finite sets within finite fields and rings, extending classical geometric problems to algebraic structures.

Findings

Derived conditions for the expected number of dot product chains

Extended Erd
53s' problem to finite fields and rings

Provided probabilistic estimates for chain counts

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) $(\alpha_1,\ldots,\alpha_k)$, 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 $1\leq j \leq k$.