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

Step 3 of 5: The non-sticky case: hunting for a Frostman violation

In plain words

If the thickened Kakeya set $E=\bigcup T$ genuinely had dimension exactly $d$ everywhere at every scale, then a ball $B(x,\rho)$ around any point $x\in E$ should contain about $(\rho/\delta)^d\delta^3$ worth of volume from $E$ — no more, no less, this being the natural 'Frostman' density of a $d$ -dimensional set thickened at scale $\delta$ . Wang and Zahl's key idea is to show that in the problematic non-sticky case, some ball must violate this upper density bound, containing noticeably more than the expected amount. Once such a 'Frostman violation' is found at some intermediate scale $\rho$ , a short covering argument upgrades it, all by itself, into the desired gain $K(d+\alpha)$ for the whole set $E$ — reducing the entire remaining problem to manufacturing one such violation.

|E\cap B(x,\rho)| \gtrsim (\rho/\delta)^{d}\delta^{3-\alpha} \ \ (\text{a Frostman violation}) \ \Longrightarrow\ |E|\gtrsim \rho^{-d}\cdot(\rho/\delta)^d\delta^{3-\alpha} = \delta^{3-d-\alpha}

Detailed analysis

Formally, a Frostman violation at scale $\rho$ means a ball where the local density exceeds the $d$ -dimensional prediction by a factor $\delta^{-\alpha}$ for the target gain $\alpha$ . Once found, a standard pigeonholing/covering argument (present already in the sticky-case papers of Katz–Łaba–Tao and refined by Wang–Zahl) upgrades a single such local violation into the global statement $K(d+\alpha)$ : cover $E$ by $\sim\rho^{-d}$ balls of radius $\rho$ centered on $E$ , note the violating ball alone already contributes more than the $K(d)$ -predicted share of the total volume, and a short volume-counting argument shows the total must exceed $\delta^{3-d-\alpha}$ . What remains, across the rest of the proof, is to locate this violation whenever the tube configuration is non-sticky.