Joints formed by lines and a $k$-plane, and a discrete estimate of Kakeya type

In 2008, Zeev Dvir solved the finite-field Kakeya conjecture using the polynomial method. His remarkably short argument led to a burst of activity, which in turn led to the solution of a number of other problems using the polynomial method, a notable example of which was the solution by Larry Guth and Nets Katz of the Erdős distinct-distances problem in 2010. Another result of Guth and Katz, proved in 2008, was the solution to a problem known as the joints conjecture. If $\mathbb F$ is a field and $\mathcal L$ is a set of lines in $\mathbb F^3$, then a joint in $\mathbb F^3$ is a point contained in set of three lines from $\mathcal L$ that have linearly independent directions. Given a set of $n$ lines, one can ask how many joints there can be. If one takes all the axis-aligned lines through a $(n/3)^{1/2}\times (n/3)^{1/2}\times (n/3)^{1/2}$ grid, one obtains a bound of $cn^{3/2}$, and the conjecture proved by Guth and Katz was that $O(n^{3/2})$ was an upper bound when $\mathbb F=\mathbb R$.

This paper considers the following higher-dimensional generalization of the joints problem. Let $k_1+\dots+k_d=n$, where all the numbers concerned are positive integers, let $\mathcal P_1,\dots,\mathcal P_d$ be families of affine subspaces of $\mathbb F^n$, with each $P_i\in\mathcal P_i$ of dimension $k_i$, and define a multijoint to be a point $x\in\mathbb F^n$ such that there is a sequence of affine subspaces $P_1,\dots,P_d$ satisfying the following three conditions.

1. For each $i$, $P_i\in\mathcal P_i$.

2. $x\in\bigcap_{i=1}^dP_i$.

3. $P_1+\dots+P_d=\mathbb F^n$.

By considering grids as in the case of joints in $\mathbb F^3$, one arrives naturally at the conjecture that the number of multijoints is at most

$$|\mathcal P_1|^{1/(d-1)}\dots|\mathcal P_d|^{1/(d-1)}.$$

Note that if $d=3$, $k_1=k_2=k_3=1$, and $\mathcal P_1=\mathcal P_2=\mathcal P_3$, then we recover the joints conjecture.

One of the main results of this paper is the first proof of a case of the multijoints conjecture where not all the $k_i$ are equal to 1: it proves the result when all but one of the $k_i$ are equal to 1. This result was proved independently by Yu and Zhao, and a few months later, while this paper was being prepared for publication, the whole conjecture was proved by Tidor, Yu and Zhao.

The second main result of the paper concerns lines in $\mathbb R^3$. Given a set $\mathcal L$ of lines, let $J$ be the set of joints determined by $\mathcal L$, and for each point $x\in\mathbb R^3$ let $L(x)$ be the number of lines in $\mathcal L$ that contain $x$. The authors prove the inequality

$$\sum_{x\in J}L(x)^{3/2}\ll |\mathcal L|^{3/2},$$

which is related to the Kakeya maximal problem. They provide a simple example to show that the inequality fails for lines in $\mathbb F_p^3$: thus, this is a specifically Euclidean result, and they use (as they must) specifically Euclidean methods to prove it. Since $L(x)\geq 3$ for every joint $x$, this result trivially implies the Guth-Katz bound, but since a point may be contained in many lines from $\mathcal L$, it is a considerable strengthening. They also provide a structural description of the set of joints that lie in at most $O(|\mathcal L|^{1/2})$ lines from $\mathcal L$. The reason that the threshold $|\mathcal L|^{1/2}$ is a natural one is connected to a threshold that appears in the Szemerédi-Trotter theorem. A consequence of that theorem is that the number of points $x$ for which $k\leq L(x)\leq 2k$ is $O(|\mathcal L|^2/k^3)$ when $k\leq|\mathcal L|^{1/2}$ and $O(|\mathcal L|/k)$ when $k\geq|\mathcal L|^{1/2})$.