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

Group the $\delta\times\delta\times 1$ thin tubes into $\rho\times\rho\times1$ fat tubes at an intermediate scale $\delta\ll\rho\ll 1$. In the most favorable ('sticky') case, the thin tubes cluster inside fat tubes as tightly as the direction-separation allows, so each fat tube really does behave like a rescaled copy of the original configuration. Assuming $K(d)$, applying it once at the coarse scale $\rho$ bounds how many fat tubes can pile on a point ($\mu_{\text{fat}}$), and applying it again after zooming into a single fat tube (rescaled by $1/\rho$) bounds how many thin tubes pile up inside it ($\mu_{\text{fine}}$). A point can only lie in a thin tube if it lies in some fat tube containing it and then in that thin tube within the fat tube, so the overall multiplicity is at most the product $\mu_{\text{fat}}\mu_{\text{fine}}$ — and multiplying the two power-law bounds from applying $K(d)$ twice exactly reproduces $K(d)$ again, with room to spare for a genuine gain $K(d+\alpha)$.