Bounds on distinct and repeated dot product trees

TL;DR

This paper investigates the number of distinct and repeated dot product configurations in large finite point sets, providing new bounds and demonstrating the prevalence of certain dot product patterns in geometric arrangements.

Contribution

It introduces new lower bounds on the number of distinct dot product sets for weighted trees in finite point sets and shows the frequent occurrence of specific dot product patterns.

Findings

New lower bounds on distinct dot product sets

Existence of many repeated dot product configurations

Narrowing the gap between known bounds

Abstract

We study questions inspired by Erd\H os' celebrated distance problems with dot products in lieu of distances, and for more than a single pair of points. In particular, we study point configurations present in large finite point sets in the plane that are described by weighted trees. We give new lower bounds on the number of distinct sets of dot products serving as weights for a given type of tree in any large finite point set. We also as demonstrate the existence of many repetitions of some special sets of dot products occurring in a given type of tree in different constructions, narrowing gap between the best known upper and lower bounds on these configurations.