Improved Bounds for Pencils of Lines

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.