Complete the higher-dimensional wave-equation divergence-set conjecture

Determine the exact supremum of the Hausdorff dimensions of divergence sets for pointwise convergence of the wave equation in dimensions n≥4 throughout the range 1/2<s<(n+1)/4, thereby resolving the remaining cases of the Barceló–Bennett–Carbery–Rogers conjecture not settled by the stated estimates.

Background

The paper studies pointwise convergence to the initial data for solutions of the wave equation with Sobolev initial data in R{n+1}. The quantity \mathcal{D}_n(s) is the supremum of the Hausdorff dimensions of exceptional divergence sets over all admissible initial data.

The introduction states that the conjectured sharp bound had previously been unresolved for n≥4 in the range 1/2<s<(n+1)/4. The paper improves the upper bound in this range and proves the conjectured value only for n=4 and 1/2<s≤11/20, so the broader higher-dimensional problem remains unresolved.

References

Therefore, prior to this work the conjecture was open in the range $1/2 < s < (n+1)/4$ when $n \geq 4$.

— The divergence set for the wave equation in higher dimensions  (2609.19691 - Du et al., 17 Sep 2026) in Section 1, Introduction

It is not yet known whether Stein's maximal principle applies in the fractal setting, but possibly the mass transference technique from \ could be applied to the counterexample in Proposition~\ref{counterexample}, with some work, to yield lower bounds for $\mathcal{D}_n(s)$ corresponding to Proposition~\ref{counterexample}, which would disprove the conjecture `divergencesup` when $n \geq 6$.

— The divergence set for the wave equation in higher dimensions  (2609.19691 - Du et al., 17 Sep 2026) in Section 2, final paragraph before Section 3