Line-Plane Incidence Bound in ℝ⁴

2023/12/22 by Cheng, Chao
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.14986

Abstract

We consider an incidence problem in ℝ⁴ which asks, for a set of L lines and a set of S planes in general position, what the maximum number of line-plane incidences is. A line-plane incidence is defined as a point where a line and a plane intersect. We prove that, when the lines and planes are in a truly 4-dimensional configuration such that no more than L^(1)/(2)+ε lines are contained in any 2-dimensional surface of degree at most D and no more than S^(1)/(2)+ε 2-planes are contained in any 3-dimensional hypersurface of degree at most D, and if L^1/2 ≪ S ≪ L, then for a constant D>1 and an ε>0 there exists a non-trivial upper bound for incidences between lines and planes: L^{(3)/(4)+(1)/(2)ε}S + LS^(1)/(2)+ε. We also prove several supporting lemmas.

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