This post is about a distinct distances problem which I think is quite interesting. As we shall see, solving this problem would also lead to significant progress in characterizing point sets with a small number of distinct distances. Recall that a perpendicular bisector of two points is the line that consists of the points satisfying (where is the distance between and ). For brevity, we will simply refer to these as bisectors.
We say that a set of points in has no repeated bisectors if for every four distinct points the bisector of and is not identical to the bisector of and (that is, the configuration in the above figure is forbidden).
Conjecture. Let be a set of points in with no repeated bisectors. Then for any the points of determine distinct distances, .
Update July 2017: The conjecture is false, as pointed out by Cosmin Pohoata. A known construction is a counterexample, and I should be ashamed for not observing this on my own. Erdős, Füredi, Pach, and Ruzsa constructed a set of points in with distinct distances for any and no four points on a common circle. Having a repeated bisector requires having four points on a circle, so there are no repeated bisectors. Oh well…
For some initial intuition, consider the point sets that determine a small number of distinct distances. I am only aware of two types of configurations that yield distinct distances: the vertices of a regular -gon and lattices.
We can also play a bit with these examples. For example, by taking two regular -gons whose vertices are on two concentric circles, or by taking evenly spaced points on a line (that is, an lattice). All of these examples have a significantly large amount of repeated bisectors.
So far I only managed to find a set with no repeated bisectors and distinct distances. From the other direction, I can only show that every such set determines distinct distances. This is unfortunate, since a stronger bound for this problem would lead to a breakthrough in characterizing planar point sets that determine a small number of distinct distances. Below I describe how this can be done, and also how to obtain simple lower and upper bounds for this problem.