Algebraic Methods in Discrete Analogs of the Kakeya Problem

Details Algebraic Methods in Discrete Analogs of the Kakeya Problem Algebraic Methods in Discrete Analogs of the Kakeya Problem

2008-12-04

arxiv.org/abs/0812.1043v1

Mathematics

Combinatorics

12 pages

Scientific paper

We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, 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 $N^2$ lines in ${\Bbb R}^3$ 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 $\Omega(N^3)$. Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.

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