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Improved bounds for pencils of lines


Authors: Oliver Roche-Newton and Audie Warren
Journal: Proc. Amer. Math. Soc. 149 (2021), 805-815
MSC (2010): Primary 52C10, 11B30
DOI: https://doi.org/10.1090/proc/14641
Published electronically: December 8, 2020
MathSciNet review: 4198085
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Abstract:

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is $O(n^{11/6})$, improving a result of Chang and Solymosi.

We also consider constructions for this problem. Alon, Ruzsa, and Solymosi constructed an arrangement of four non-collinear $n$-pencils which determine $\Omega (n^{3/2})$ four-rich points. We give a construction to show that this is not tight, improving this lower bound by a logarithmic factor. We also give a construction of a set of $m$ $n$-pencils, whose centres are in general position, that determine $\Omega _m(n^{3/2})$ $m$-rich points.


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Additional Information

Oliver Roche-Newton
Affiliation: Johann Radon Institute for Compuational and Applied Mathematics (Ricam), Linz, Austria
MR Author ID: 949899
Email: o.rochenewton@gmail.com

Audie Warren
Affiliation: Johann Radon Institute for Compuational and Applied Mathematics (Ricam), Linz, Austria
Email: audie.warren@oeaw.ac.at

Received by editor(s): May 28, 2018
Received by editor(s) in revised form: January 18, 2019
Published electronically: December 8, 2020
Additional Notes: Both authors were supported by the Austrian Science Fund (FWF) Project P 30405-N32
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2020 American Mathematical Society