On almost-equidistant sets. II (Q2415075)

Summary: A set in \(\mathbb{R}^d\) is called almost-equidistant if for any three distinct points in the set, some two are at unit distance apart. First, we give a short proof of the result of \textit{K. Bezdek} and \textit{Z. Lángi} [Period. Math. Hung. 39, No. 1--3, 139--144 (1999; Zbl 0965.51006)] claiming that an almost-equidistant set lying on a \((d-1)\)-dimensional sphere of radius \(r\), where \(r<1/\sqrt{2}\), has at most \(2d+2\) points. Second, we prove that an almost-equidistant set \(V\) in \(\mathbb{R}^d\) has \(O(d)\) points in two cases: if the diameter of \(V\) is at most 1 or if \(V\) is a subset of a \(d\)-dimensional ball of radius at most \(1/\sqrt{2}+cd^{-2/3}\), where \(c<1/2\). Also, we present a new proof of the result of \textit{A. Kupavskii}, \textit{N. H. Mustafa} and \textit{K. J. Swanepoel} [``Bounding the size of an almost-equidistant set in Euclidean space'', Preprint, \url{arXiv:1708.01590}] that an almost-equidistant set in \(\mathbb{R}^d\) has \(O(d^{4/3})\) elements. \par For Part I see [the author, Linear Algebra Appl. 563, 220--230 (2019; Zbl 1407.52018)].