Distinct distances on algebraic curves in the plane

Abstract

Let S be a set of n points in R2 contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least cdn4/3, unless C contains a line or a circle. We also prove the lower bound c'd min{m2/3n2/3, m2, n2} for the number of distinct distances between m points on one irreducible plane algebraic curve and n points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer, and Solymosi in [18].