The divergence set for the wave equation in higher dimensions

Abstract: It is shown that if $u$ solves the wave equation in $\mathbb{R}<sup>{4+1}$ with initial data $u(\cdot,0) = u_0(\cdot) \in H<sup>s$ and $\partial_tu(\cdot,0) = u_1(\cdot ) \in H<sup>{s-1}$, where $0.5 &lt; s \leq 0.55$, then $u(x,t) \to u_0(x)$ and $\partial_tu(x,t) \to u_1(x)$ pointwise as $t \to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in $\mathbb{R}<sup>{n+1}$ is obtained for $n \geq 4$ and $1/2 < s < n/4$.