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 \begin{equation*} |A| \leq {d+s \choose s}. \end{equation*} 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.