Bounding the size of an almost-equidistant set in Euclidean space

TL;DR

This paper establishes an upper bound on the size of almost-equidistant point sets in Euclidean space, showing they cannot grow faster than a constant times d^{4/3}.

Contribution

It provides the first non-trivial upper bound on the size of almost-equidistant sets in high-dimensional Euclidean spaces.

Findings

Almost-equidistant sets in  have size O(d^{4/3})

The bound improves understanding of geometric configurations in high dimensions

Sets cannot be arbitrarily large if they are almost equidistant

Abstract

A set of points in d-dimensional Euclidean space is almost equidistant if among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in $\mathbb{R}^d$ has cardinality $O(d^{4/3})$.