A note on distinct distances

We show that, for a constant-degree algebraic curve γ in aD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2]. We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in aD, there exists a subset S AŠ' P of size at least Ω(n4/(9+12(d-1))), such that S spans distinct distances. This improves the earlier bound of Ω(n1/(3d)) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4]. Both results are consequences of a common technical tool.