Bounds on point configurations determined by distances and dot products

2020/11/30 by Gunter, Slade, Palsson, Eyvi, Rhodes, Ben +1
#51A20 #52C10 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2011.15055

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.

Related

• Dot product chains
• Bounds on distinct and repeated dot product trees
• Angle chains and pinned variants
• On the number of dot product chains in finite fields and rings
• On the unit distance problem