The constant of point–line incidence constructions

Martin Balko; Adam Sheffer; Ruiwen Tang

doi:10.1016/j.comgeo.2023.102009

The constant of point–line incidence constructions

Martin Balko, Adam Sheffer, Ruiwen Tang

Research output: Contribution to journal › Article › peer-review

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)≥(c+o(1))|P|^2/3|L|^2/3, with c≈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.

Original language	English
Article number	102009
Journal	Computational Geometry: Theory and Applications
Volume	114
DOIs	https://doi.org/10.1016/j.comgeo.2023.102009
State	Published - 1 Oct 2023
Externally published	Yes

Keywords

Integer lattice
Point-line incidence
Szemerédi–Trotter theorem
Totient function

ASJC Scopus subject areas

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2023.102009

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title = "The constant of point–line incidence constructions",

abstract = "We study a lower bound for the constant of the Szemer{\'e}di–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≈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 o}s's construction.",

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AU - Balko, Martin

AU - Sheffer, Adam

AU - Tang, Ruiwen

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AB - 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)≥(c+o(1))|P|2/3|L|2/3, with c≈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.

KW - Integer lattice

KW - Point-line incidence

KW - Szemerédi–Trotter theorem

KW - Totient function

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DO - 10.1016/j.comgeo.2023.102009

M3 - Article

AN - SCOPUS:85154595901

SN - 0925-7721

VL - 114

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 102009

ER -

The constant of point–line incidence constructions

Abstract

Keywords

ASJC Scopus subject areas

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