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

Step 4 of 5: Structure theorem: tubes organize into 'grains'

In plain words

Rather than tiling $E$ crudely by $\delta\times\delta\times\delta$ cubes, Wang and Zahl show (adapting Guth's polynomial-method approach to graininess, since the stickiness-based route of Katz–Łaba–Tao is unavailable here) that the tubes inside a fat tube $T_\rho$ organize into essentially disjoint 'grains' — flat rectangular prisms of some intermediate, a priori unknown dimensions $\delta\times \rho c\times c$ , taken as large as possible. Replacing the wasteful fat-tube multiplicity $\mu_{\text{fat}}$ from Step 2 by the multiplicity of these tighter-fitting grains, $\mu_{\text{coarse}}$ , gives a strictly better multiplicity inequality $\mu\lesssim\mu_{\text{coarse}}\mu_{\text{fine}}$ : if, after rescaling, the grains behave like an ordinary (or sparser) Kakeya configuration, $K(d)$ already bounds $\mu_{\text{coarse}}\lesssim \rho^{-d}$ and a companion bound on $\mu_{\text{fine}}$ closes the induction outright, exactly as in the sticky case.

\bigcup_{T\subset T_\rho} T = \bigsqcup(\text{grains of dimension } \delta\times \rho c\times c),\qquad \mu \lesssim \mu_{\text{coarse}}\cdot\mu_{\text{fine}}, \quad \mu_{\text{coarse}} \lesssim \rho^{-d}

Detailed analysis

This step (Wang–Zahl 2025, §5, an adaptation of the 'graininess' technology introduced by Guth via the polynomial method) is the technical heart of the non-sticky case. Because Katz–Łaba–Tao's original stickiness-based graininess reduction needs an 'x-ray estimate' unavailable in this generality, Wang and Zahl instead run a polynomial partitioning argument to find, inside each fat tube $T_\rho$ , the coarsest possible decomposition of the enclosed thin tubes into flat disjoint prisms. The multiplicity inequality $\mu\lesssim\mu_{\text{coarse}}\mu_{\text{fine}}$ this yields is strictly sharper than the naive $\mu\lesssim\mu_{\text{fat}}\mu_{\text{fine}}$ of Step 2 whenever the grains are genuinely smaller than the fat tube itself, since $\mu_{\text{coarse}}$ then measures overlap of the grains rather than of the much larger, wasteful fat tubes.

Common mistake. The grain dimensions

\rho c,c

are not fixed in advance and could in principle be degenerate: if

\rho\approx c\approx 1

, the grains are enormous

\delta\times 1\times 1

slabs and

E

looks like a union of planar sheets — here the classical Córdoba

L^2

argument from the planar case (Step 3 of the

\mathbb{R}^2

proof) already suffices and this case is easy. The genuinely hard regime is when the rescaled grains do not behave like a Kakeya or sub-Kakeya configuration at all, which forces the structure theorem of the next step.