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 $I(P,L)\ge (c+o(1\left)\right)\left|P{|}^{2/3}\right|L{|}^{2/3}$, 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 Erdős's construction.