Pairs of dot products in finite fields and rings

TL;DR

This paper establishes bounds on the number of triples in finite fields or rings that determine specific pairs of dot products, advancing understanding of geometric configurations in algebraic structures.

Contribution

It provides new bounds on the count of triples with prescribed dot product pairs in finite fields and rings, extending geometric combinatorics in algebraic settings.

Findings

Bounds on the number of triples with given dot product pairs

Results applicable to finite fields and rings

Enhanced understanding of geometric configurations

Abstract

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 \[\left\{(u,v,w)\in E \times E \times E : u\cdot v = \alpha, u \cdot w = \beta \right\}\] for units $\alpha$ and $\beta$.