Incidences between points and lines in R^4

Abstract

We show that the number of incidences between $m$ distinct points and $n$ distinct lines in ${\mathbb{R}}^4$ is $O\left(2^{c\sqrt{\log m}}(m^{2/5}{n}^{4/5}+m)+m^{1/2}{n}^{1/2}{q}^{1/4}+m^{2/3}{n}^{1/3}{s}^{1/3}+n\right)$, for a suitable absolute constant $c$, provided that no 2-plane contains more than $s$ input lines, and no hyperplane or quadric contains more than $q$ lines. The bound holds without the factor $2^{c\sqrt{\log m}}$ when $m\le n^{6/7}$ or $m\ge n^{5/3}$. Except for this factor, the bound is tight in the worst case.

References in corpus (2)

• On the Erdos distinct distance problem in the plane
• Simple Proofs of Classical Theorems in Discrete Geometry via the Guth--Katz Polynomial Partitioning Technique