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 ℝ^d, which determines s distinct distances, then |A| ≤ (d+s/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.