Distinct Distances Between a Line and Strip

Abstract

We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances between points on two lines or curves. Unlike these existing problems, this new variant only makes sense if the points satisfy an additional spacing condition. Our work can also be seen as an exploration of the proximity technique that was recently introduced by Solymosi and Zahl. This technique lies at the heart of our analysis.