The constant of point-line incidence constructions

Abstract

We study a lower bound for the constant of the Szemeredi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies I(P, L) = (c + o(1))|P|(2/3)|L|(2/3), with c similar to 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 Erdos's construction. (c) 2023 Elsevier B.V. All rights reserved.

Keywords

• Point-line incidence
• Szemeredi-Trotter theorem
• Totient function
• Integer lattice