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

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 $$|A|\le \left(\genfrac{}{}{0ex}{}{d+s}{s}\right).$$ 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.