Barceló–Bennett–Carbery–Rogers conjecture on wave-equation divergence sets

Let $d\geq 3$ and, for $s>0$, define

$$\mathcal D_d(s):=\sup_{f\in H^s(\mathbb R^d)}\dim_H\mathfrak D_d(f),$$

where

$$\mathfrak D_d(f):=\{x:e^{it\sqrt{-\Delta}}f(x)\not\to f(x)\text{ as }t\to 0\}$$

and $\dim_H$ denotes Hausdorff dimension. Barceló–Bennett–Carbery–Rogers conjecture.

$$\mathcal D_d(s)=\begin{cases}d-2s,&1\leq s\leq \frac d2,\d+2-4s,&\frac12<s\leq 1.\end{cases}$$

The quantity $\mathcal D_d(s)$ measures the largest possible Hausdorff dimension of the divergence set for initial data in $H^s(\mathbb R^d)$. The formula is known in dimension two, while the higher-dimensional case stated here remains open in the supplied source.