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- W2803607446 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." @default.
- W2803607446 created "2018-06-01" @default.
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- W2803607446 date "2018-05-23" @default.
- W2803607446 modified "2023-09-27" @default.
- W2803607446 title "Improved Bounds for Pencils of Lines" @default.
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