The constant of point-line incidence constructions

Abstract

We study a lower bound for the constant of the Szemerédi-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.