On lines and Joints

Abstract

Let $L$ be a set of $n$ lines in ${\mathbb{R}}^d$, for $d\ge 3$. A {\em joint} of $L$ is a point incident to at least $d$ lines of $L$, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of $L$ is $\Theta (n^{d/(d-1)})$. For $d=3$, this is a considerable simplification of the orignal algebraic proof of Guth and Katz~\cite{GK}, and of the follow-up simpler proof of Elekes et al. \cite{EKS}.

References in corpus (3)

• On Lines, Joints, and Incidences in Three Dimensions
• Algebraic Methods in Discrete Analogs of the Kakeya Problem
• On the Multilinear Restriction and Kakeya conjectures