Improved Bounds for Pencils of Lines

2018/05/23 by Roche-Newton, Oliver, Warren, Audie · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.09188

Abstract

We consider a question raised by Rudnev: given four pencils of n concurrent lines in \mathbb R², 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 Ω(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 Ω_m(n^3/2) m-rich points.

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