Sharp Szemer\'edi-Trotter Constructions in the Plane

TL;DR

This paper introduces a novel family of sharp point-line incidence examples in the plane, breaking from traditional lattice-based constructions, and applies these findings to the inverse Loomis-Whitney problem.

Contribution

It provides the first non-lattice-based sharp examples for the Szemerédi-Trotter theorem and connects these to the inverse Loomis-Whitney problem.

Findings

New non-lattice sharp examples for Szemerédi-Trotter theorem

Application to discrete inverse Loomis-Whitney problem

Advancement in understanding point-line incidences

Abstract

We present a new family of sharp examples for the Szemer\'edi-Trotter theorem. These are the first examples not based on a rectangular lattice. We also include an application to the discrete inverse Loomis-Whitney problem.