Discrete Analysis Seminar

We study sharp examples of the Szemerédi--Trotter bound: constructions of points and lines in the plane with maximum number of incidences. In joint work with Adam Sheffer, we prove structural properties of such constructions when the number of points and lines are equal, and the points form a Cartesian product. We show there exist many families of parallel lines or many families of concurrent lines.

We also introduce the first infinite family of sharp Szemerédi-Trotter constructions, and in joint work with Larry Guth, the first such family with no underlying lattice structure, which yield sharp discrete Loomis-Whitney constructions as a corollary.

In the work with Adam Sheffer, our techniques are based on the concept of line energy. Rudnev and Shkredov recently introduced this concept and showed how it is connected to point--line incidences. We prove that their bound is tight up to sub-polynomial factors.