Dot product chains

TL;DR

This paper investigates the maximum number of point sequences with prescribed dot products in a large finite set, extending classical geometric problems to a new algebraic setting.

Contribution

It provides new bounds on the number of point tuples with specified dot product relations, advancing understanding of geometric configurations in finite sets.

Findings

Established upper bounds on dot product chains.

Derived lower bounds demonstrating the existence of many such chains.

Extended classical geometric problems to algebraic dot product sequences.

Abstract

We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of nonzero dot products $(\alpha_1,\ldots,\alpha_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(A_1,\dots, A_{k+1})\in E^{k+1}$ satisfying $A_j \cdot A_{j+1}=\alpha_j$ for every $1\leq j \leq k$.