  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 \[ s > \frac{d}{2(d+2)}\left( d+2-\alpha \right) \]
  is sufficient for the solution corresponding to every datum in $H^s(\mathbb
T^d)$ to converge to such datum $\alpha$-almost everywhere.
