Crescent configurations in three dimensions

Abstract

A crescent configuration of $n$ points in $\mathbb R^d$ is a set in general position whose distinct pairwise distances have multiplicities $1,\ldots,n-1$. We construct such a configuration in $\mathbb R^3$ for every $n\geq3$, giving a uniform bound of three for the minimum dimension. The proof uses equally spaced parameters on a short circular helix and Rolle's theorem. We also give explicit integer-coordinate configurations with $O(n\log n)$ bits per coordinate, and prove that every finite initial segment of the sequence $(\cos j,\sin j,2j)_{j\geq0}$ of length at least three is a crescent configuration.