Algebraic methods in discrete analogs of the Kakeya problem

We prove the joints conjecture, showing that for any N lines in R 3 , there are at most O ( N 3 2 ) points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line intersects a...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2010-12, Vol.225 (5), p.2828-2839
Main Authors: Guth, Larry, Katz, Nets Hawk
Format: Article
Language:English
Subjects:
Bezout's theorem
Dvir argument
Incidence problem
Joints
Kakeya problem
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Summary:We prove the joints conjecture, showing that for any N lines in R 3 , there are at most O ( N 3 2 ) points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 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 ( N 3 ) . Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2010.05.015