Algebraic methods in discrete analogs of the Kakeya problem

By Larry Guth, Nets Hawk Katz. 2008

137 citations recorded by the source corpus.

Abstract 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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a1c6b49d8dcaf3a15e591bf403ac145ee7927f3f

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In Applied Mathematics

1. J. Kollár
2. J. Sparks

In Discrete Mathematics and Combinatorics

1. J. Kollár

In Geometry and Topology

1. J. Kollár
2. O. Fujino
3. Christopher D. Hacon
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6. J. Sparks
7. Hiromu Tanaka
8. Chenyang Xu

Citations in the corpus

• Szemer\'edi--Trotter-type theorems in dimension 3 (2014, background): “; see Remark 15 for details. For the applications in [GK15] the relevant value of c is ≤3.5. 7 (Comparisonwith previous results). The idea of using algebraicsurfaces to attack such problems is due to [GK10]. Our main observation is that the arithmetic genus of a line configuration provides a very effic”