A note on distinct distances

Abstract: We show that, for a constant-degree algebraic curve $\gamma$ in $\mathbb{R}<sup>D$, every set of $n$ points on $\gamma$ spans at least $\Omega(n<sup>{4/3})$ distinct distances, unless $\gamma$ is an {\it algebraic helix} (see Definition 1.1). This improves the earlier bound $\Omega(n<sup>{5/4})$ of Charalambides [Discrete Comput. Geom. (2014)]. We also show that, for every set $P$ of $n$ points that lie on a $d$-dimensional constant-degree algebraic variety $V$ in $\mathbb{R}<sup>D$, there exists a subset $S\subset P$ of size at least $\Omega(n<sup>{\frac{4}{9+12(d-1)}})$, such that $S$ spans $\binom{|S|}{2}$ distinct distances. This improves the earlier bound of $\Omega(n<sup>{\frac{1}{3d}})$ of Conlon et al. [SIAM J. Discrete Math. (2015)]. Both results are consequences of a common technical tool, given in Lemma 2.7 below.