Namboodiri Lecture 2: the Distinct Distance Problem

Last lecture we introduced the cutting method, and we sketched the proof of Theorem 2. We can pose variations on Question 1 by replacing lines with other curves in the plane. What happens for circles? Unit circles? Parabolas? Ellipses? These are four major open problems of incidence geometry! Let us try to explain why these problems are hard, or at least why they cannot be solved by the same method that worked for lines. The short answer is that the cutting method gives some upper bounds in all four of these problems, but the bounds do not match any examples. We focus on the case of unit circles, which is the most intensively studied. Suppose that Γ is a set of L unit circles. The bound for |Pr(Γ)| from the cutting method is exactly the same as for lines. The cutting method combines a topological argument and a double counting argument, and both of these arguments work just the same for lines and for unit circles. In both cases it is easy to make a stars example with Lr−1 r-rich points. But the grid example does not work as well for circles. The problem is that a circle cannot contain very many lattice points. Given an S×S grid of points, there are lines that contain S grid points, but the number of grid points in any circle is O(S ) for any > 0. We summarize these bounds and examples in the following table: