A remark on sets with few distances in $\mathbb{R}^{d}$

TL;DR

This paper provides a new, simplified proof of a classical bound on the size of point sets in Euclidean space with few distinct distances, using quadratic forms and additive combinatorics techniques.

Contribution

It introduces a novel proof method combining Sylvester's Law of Inertia with the Croot-Lev-Pach Lemma to establish the distance set bound.

Findings

Simplified proof of the Bannai-Bannai-Stanton theorem

Bound |A| ≤ (d+s choose s) for sets with s distances in R^d

Method bridges quadratic forms and additive combinatorics

Abstract

A celebrated theorem due to Bannai-Bannai-Stanton says that if $A$ is a set of points in $\mathbb{R}^{d}$, which determines $s$ distinct distances, then $$|A| \leq {d+s \choose s}.$$ In this note, we give a new simple proof of this result by combining Sylvester's Law of Inertia for quadratic forms with the proof of the so-called Croot-Lev-Pach Lemma from additive combinatorics.