Bounds on point configurations determined by distances and dot products

TL;DR

This paper investigates bounds on the number of point subsets satisfying specific distance and dot product relations within large finite sets, extending Erdős' unit distance problem and presenting new general bounds.

Contribution

It introduces new, more general bounds on point configurations related to distances and dot products, expanding recent research in the area.

Findings

Derived bounds for subsets with prescribed distance relations

Established bounds for subsets with prescribed dot product relations

Surveyed recent advances in point configuration problems

Abstract

We study a family of variants of Erd\H os' unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, we look for bounds on how many subsets of $k$ points satisfy a set of relationships between point pairs based on distances or dot products. We survey some of the recent work in the area and present several new, more general families of bounds.