Algebraic Methods in Discrete Analogs of the Kakeya Problem

Abstract

We prove the joints conjecture, showing that for any $N$ lines in ${\mathbb{R}}^3$, there are at most $O(N^{\frac{3}{2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines in ${\mathbb{R}}^3$ 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.

12 pages

References in corpus (2)

• On a question of Bourgain about geometric incidences
• The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture

Cited by in corpus (5)

• Counting joints with multiplicities
• On Lines, Joints, and Incidences in Three Dimensions
• On lines and Joints
• Incidences in Three Dimensions and Distinct Distances in the Plane
• The joints problem for matroids