Algebraic Methods in Discrete Analogs of the Kakeya Problem

2008/12/04 by Larry Guth, Guth, Larry, Nets Hawk Katz +1 · 1 citation
Mathematics · #14A25 #52C10 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.0812.1043

openalex publication_date 2008/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the joints conjecture, showing that for any N lines in \Bbb R³, there are at most O(N^{3 \over 2}) points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N² lines in \Bbb R³ so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N³). Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.

Cited by

• On Lines, Joints, and Incidences in Three Dimensions

Related

• Discrete analogues of Kakeya problems
• Counting multijoints
• The Kakeya Conjecture: where does it come from and why is it important?
• A finite version of the Kakeya problem
• On Lines, Joints, and Incidences in Three Dimensions