Structural Szemer\'edi-Trotter for Lattices and their Generalizations

TL;DR

This paper characterizes point-line incidences in lattice sections, advancing the structural Szemerédi-Trotter problem, and explores generalizations involving Cartesian products and arithmetic progressions.

Contribution

It provides a complete characterization for lattice point-line incidences and partial results for broader configurations, using multiplicative energy techniques.

Findings

Characterization of $ heta(n^{4/3})$ incidences in lattice sections

Ruling out concurrent lines in certain Cartesian product configurations

Analysis of generalized arithmetic progressions in point-line incidences

Abstract

We completely characterize point--line configurations with $\Theta(n^{4/3})$ incidences when the point set is a section of the integer lattice. This can be seen as the main special case of the structural Szemer\'edi-Trotter problem. We also derive a partial characterization for several generalizations: (i) We rule out the concurrent lines case when the point set is a Cartesian product of an arithmetic progression and an arbitrary set. (ii) We study the case of a Cartesian product where one or both sets are generalized arithmetic progression. Our proofs rely on deriving properties of multiplicative energies.