Improved bounds for pencils of lines,Proceedings of the American Mathematical Society

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Improved bounds for pencils of lines
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-12-08 , DOI: 10.1090/proc/14641
Oliver Roche-Newton , Audie Warren

Abstract:We consider a question raised by Rudnev: given four pencils of

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

-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

-pencils, whose centres are in general position, that determine $\Omega _m(n^{3/2})$

-rich points.

[1] N. Alon, I. Ruzsa, and J. Solymosi, Sums, products and ratios along the edges of a graph, preprint, arXiv:1802.06405, 2018.
[2]

中文翻译：

改进了直线铅笔的边界

摘要：我们考虑鲁德涅夫提出的一个问题：给定四根

同时存在的铅笔，且铅笔的四个中心不共线，那么四根线相交的点集的最大可能大小是多少？我们的主要结果表明，此类点的数量为，改善了Chang和Solymosi的结果。我们还考虑了此问题的构造。Alon，Ruzsa和Solymosi构建了四个非共线铅笔的排列，这些铅笔确定了四个富点。我们给出了一个结构来表明这并不严格，通过对数因子改善了该下限。我们还给出了一组-铅笔的构造，这些铅笔的中心在一般位置上，它们确定-富点。 $\ mathbb {R} ^ 2$ $O（n ^ {11/6}）$

$\ Omega（n ^ {3/2}）$

$\ Omega _m（n ^ {3/2}）$

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