Counting multijoints

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 $|J({\mathfrak{L}}_1,{\mathfrak{L}}_2,{\mathfrak{L}}_3)|\lesssim (L_1{L}_2{L}_3{)}^{1/2}$, 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.

25 pages. arXiv admin note: substantial text overlap with arXiv:1312.5436

References in corpus (1)

• Discrete analogues of Kakeya problems