Improved Bounds for Pencils of Lines

TL;DR

This paper improves the upper bounds on the number of four-line intersection points in four pencils of lines in the plane and provides new constructions that surpass previous lower bounds, advancing combinatorial geometry understanding.

Contribution

It establishes a tighter upper bound of O(n^{11/6}) for four-pencil intersection points and introduces constructions that improve lower bounds for rich point counts.

Findings

Maximum four-pencil intersection points is O(n^{11/6})

Constructed arrangements with more four-rich points than previous lower bounds

Provided general position constructions with Omega_m(n^{3/2}) m-rich points

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.