Convergence over fractals for the periodic Schr\"odinger equation

We consider a fractal refinement of Carleson’s problem for pointwise convergence of solutions to the periodic Schr"odinger equation to their initial datum. For $\alpha \in (0,d]$ and $$s<\frac{d}{2(d+1)}(d+1-\alpha ),$$ we find a function in $H^s({\mathbb{T}}^d)$ whose corresponding solution diverges in the limit $t\to 0$ on a set with strictly positive $\alpha$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that $$s>\frac{d}{2(d+2)}(d+2-\alpha )$$ is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$toconvergetosuchdatum$\alpha$-almost everywhere.