A note on distinct distances

We show that, for a constant-degree algebraic curve γ in R, every set of n points on γ spans at least Ω(n) distinct distances, unless γ is an algebraic helix (see Definition 1.1). This improves the earlier bound Ω(n) of Charalambides [2]. We also show that, for every set P of n points that lie on a d-dimensional constantdegree algebraic variety V in R, there exists a subset S ⊂ P of size at least Ω(n 4 9+12(d−1) ), such that S spans ( |S| 2 ) distinct distances. This improves the earlier bound of Ω(n 1 3d ) of Conlon et al. [4]. Both results are consequences of a common technical tool, given in Lemma 2.7 below.