The discretized three-dimensional Kakeya conjecture

Let $\varepsilon>0$. A family $L$ of lines is directionally $\delta$-separated when its line directions are separated at scale $\delta$, and let $E\subset[0,1]^3$ be a union of $\delta$-balls. For a line $\ell$, write $|\ell\cap E|$ for its length inside $E$. The discretized three-dimensional Kakeya conjecture. For every $\varepsilon>0$, there exist $c_\varepsilon>0$ and $M=M(\varepsilon)>0$ such that, for every $\delta\in(0,1)$, whenever $L$ is a family of directionally $\delta$-separated lines, $E\subset[0,1]^3$ is a union of $\delta$-balls, and $|\ell\cap E|\geq\lambda$ for every $\ell\in L$, one has

$$|E|\geq c_\varepsilon\delta^{\varepsilon}\lambda^M(\delta^2\# L).$$

After standard discretization arguments, this statement implies the three-dimensional Kakeya conjecture. It remains open; the paper establishes an epsilon improvement toward the Kakeya problem rather than proving this full estimate.