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