Generalizations of Joints Problem

Abstract

We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in ${\mathbb{R}}^6$, the number of points at which three $2$-planes intersect and span ${\mathbb{R}}^6$ is at most $C{N}^{3/2+\epsilon }$. We also get almost sharp bound on the number of joints with multiplicities. The main tools are polynomial partitioning and induction on dimension.

References in corpus (3)

• Counting joints with multiplicities
• Discrete analogues of Kakeya problems
• On the joints problem with multiplicities

Cited by in corpus (2)

• Joints of varieties
• A proof of the Multijoints Conjecture and Carbery's generalization