Convergence over fractals for the periodic Schrödinger equation

We consider a fractal refinement of Carleson's problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For $α\in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - α), \] 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 $α$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that \[ s > \frac{d}{2(d+2)}\left( d+2-α\right) \] is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$ to converge to such datum $α$-almost everywhere.