Worked solution: The three-dimensional case: every Kakeya set in $\mathbb{R}^3$ has dimension 3 (Wang–Zahl, 2025)

Fix a scale $\delta>0$ and let $\mathbb{T}$ be a family of $\delta\times\delta\times1$ tubes in $\mathbb{R}^3$ whose directions are $\delta$-separated on $S^2$ — morally $N\sim\delta^{-2}$ tubes, one per direction. The discretised Kakeya conjecture $K(d)$ asserts the uniform lower bound $\big|\bigcup_{T\in\mathbb{T}}T\big|\gtrsim\delta^{3-d}$: the union of tubes is about as large as if each carried its share of a $d$-dimensional set thickened to scale $\delta$.

For a set $E$ of genuine Hausdorff dimension $d$, the $\delta$-neighbourhood has volume $\sim\delta^{3-d}$, so $K(d)$ for all tube families is the tube-level restatement of '$\dim E\ge d$ for every Kakeya set'. Since tubes have volume $\delta^2$ and there are $\sim\delta^{-2}$ of them, disjointness would give volume $\sim1$; $K(d)$ quantifies how much overlap the geometry can possibly force.

A technicality hidden in the statement: Wang–Zahl work not just with direction-separated families but with families satisfying the 'Katz–Tao convex Wolff axioms' — non-concentration axioms stating the tubes inside any convex set have bounded multiplicity. This extra flexibility is what lets the induction on scales apply $K(d)$ to rescaled subfamilies that are no longer literally direction-separated.