Distinct distances between a collinear set and an arbitrary set of points

Abstract

We consider the number of distinct distances between two finite sets of points in Rk, for any constant dimension k≥2, where one set P1 consists of n points on a line l, and the other set P2 consists of m arbitrary points, such that no hyperplane orthogonal to l and no hypercylinder having l as its axis contains more than O(1) points of P2. The number of distinct distances between P1 and P2 is then Ωminn2∕3m2∕3,n10∕11m4∕11log2∕11m,n2,m2.Without the assumption on P2, there exist sets P1, P2 as above, with only O(m+n) distinct distances between them.