Counting joints with multiplicities

Abstract

Let $\mathfrak{L}$ be a collection of $L$ lines in ${\mathbb{R}}^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J|\lesssim L^{3/2}$ (first proved by Guth and Katz). For each joint $x\in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that ${\sum }_{x\in J}N(x{)}^{1/2}\lesssim L^{3/2}$, while in the last section we extend this result to real algebraic curves of uniformly bounded degree in ${\mathbb{R}}^3$, as well as to curves in ${\mathbb{R}}^3$ parametrised by real polynomials of uniformly bounded degree.

More details in section 4. Typos corrected. The main results are unchanged

References in corpus (4)

• On the size of Kakeya sets in finite fields
• On the Erdos distinct distance problem in the plane
• Simple Proofs of Classical Theorems in Discrete Geometry via the Guth--Katz Polynomial Partitioning Technique
• Algebraic Methods in Discrete Analogs of the Kakeya Problem

Cited by in corpus (8)

• Discrete analogues of Kakeya problems
• On the joints problem with multiplicities
• Counting joints in vector spaces over arbitrary fields
• Generalizations of Joints Problem
• Joints of varieties
• Incidence bounds on multijoints and generic joints
• Counting multijoints
• A proof of the Multijoints Conjecture and Carbery's generalization