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

Discretize: let $\mathbb{T}$ be $\approx\delta^{-2}$ tubes of dimensions $\delta\times\delta\times 1$ in $\mathbb{R}^3$, pointing in $\delta$-separated directions. Write $K(d)$ for the statement that their union always has volume $\gtrsim \delta^{3-d}$. A single tube trivially gives $K(1)$; a classical 1970s $L^2$ argument of Córdoba (the same idea behind the planar Kakeya theorem) gives $K(2)$; Wolff's 1995 'hairbrush' argument gives $K(2.5)$, and progress stalled there for three decades despite deep work by Bourgain, Katz, Tao, and others. Wang and Zahl's new preprint proves $K(3-\varepsilon)$ for every $\varepsilon>0$, which is exactly the statement that every Kakeya set in $\mathbb{R}^3$ has Hausdorff and Minkowski dimension $3$.