Counting multijoints

TL;DR

This paper establishes an upper bound on the number of multijoints formed by collections of lines and algebraic curves in three-dimensional space, extending previous results to more general curves and connecting to Kakeya-type problems.

Contribution

It proves a bound on multijoints formed by lines and extends the result to algebraic and polynomially parametrized curves in ^3, advancing the understanding of geometric incidences.

Findings

Bound |J(L_1, L_2, L_3)| \u2264 C (L_1 L_2 L_3)^{1/2}

Extension to algebraic curves of bounded degree

Extension to polynomially parametrized curves

Abstract

Let $\mathfrak{L}_1$, $\mathfrak{L}_2$, $\mathfrak{L}_3$ be finite collections of $L_1$, $L_2$, $L_3$, respectively, lines in $\mathbb{R}^3$, and $J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)$ the set of multijoints formed by them, i.e. the set of points $x \in \mathbb{R}^3$, each of which lies in at least one line $l_i \in \mathfrak{L}_i$, for all $i=1,2,3$, such that the directions of $l_1$, $l_2$ and $l_3$ span $\mathbb{R}^3$. We prove here that $|J(\mathfrak{L}_1, \mathfrak{L}_2,\mathfrak{L}_3)|\lesssim (L_1L_2L_3)^{1/2}$, and we extend our results to multijoints formed by real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as by curves in $\mathbb{R}^3$ parametrised by real univariate polynomials of uniformly bounded degree. The multijoints problem is a variant of the joints problem, as well as a discrete analogue of the endpoint multilinear Kakeya problem.