Distinct Distances Between Sets of Points on a Line and a Hyperplane in R

A variant of Erdős’s distinct distances problem considers two sets of points in Euclidean space P1 and P2, both of cardinality n, and asks whether we can find a superlinear bound on the number of distinct distances between all pairs of points with one in P1 and the other in P2. In 2013, Sharir, Sheffer, and Solymosi [8] showed a lower bound of Ω(n) when P1 and P2 are both collinear point sets in R, where the two lines defined by P1 and P2 are not orthogonal or parallel. Here, we contain P1 in a line l and P2 in a hyperplane in R. We prove that the number of distinct distances in this case has a lower bound of Ω(n) given some restrictions on l and P2.