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arXiv · 2210.04821

The constant of point-line incidence constructions

Martin Balko·Adam Sheffer·Ruiwen Tang
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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.

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Martin Balko, Adam Sheffer, Ruiwen Tang. 2022-10-10. The constant of point-line incidence constructions. https://arxiv.org/abs/2210.04821

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