The constant of point-line incidence constructions

TL;DR

This paper investigates the lower bounds of the Szemerédi-Trotter theorem's constant, improving known bounds for specific point-line configurations and analyzing Erdős's construction using properties of Euler's totient function.

Contribution

It provides a new lower bound for the constant in point-line incidence bounds and offers the first complete analysis of Erdős's construction constant.

Findings

Established a lower bound of approximately 1.27 for the incidence constant.

Improved the constant for Elekes's construction from 1 to about 1.27.

Analyzed properties of Euler's totient function related to incidence configurations.

Abstract

We study a lower bound for the constant of the Szemer\'edi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies $I({\mathcal P},{\mathcal L})\ge (c+o(1)) |{\mathcal P}|^{2/3}|{\mathcal L}|^{2/3}$, with $c\approx 1.27$. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erd\H os's construction.