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.

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• From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE
• On the Erdos distinct distance problem in the plane
• Improved bounds for incidences between points and circles
• Simple Proofs of Classical Theorems in Discrete Geometry via the Guth--Katz Polynomial Partitioning Technique
• On the number of rich lines in high dimensional real vector spaces

Cited by in corpus (2)

• Highly incidental patterns on a quadratic hypersurface in ${\mathbb{R}}^4$
• Incidences with curves in R^d