  We consider a fractal refinement of the Carleson problem for the
Schr\"odinger equation, that is to identify the minimal regularity needed by
the solutions to converge pointwise to their initial data almost everywhere
with respect to the $\alpha$-Hausdorff measure ($\alpha$-a.e.). We extend to
the fractal setting ($\alpha < n$) a recent counterexample of Bourgain
\cite{Bourgain2016}, which is sharp in the Lebesque measure setting ($\alpha =
n$). In doing so we recover the necessary condition from \cite{zbMATH07036806}
for pointwise convergence~$\alpha$-a.e. and we extend it to the range
$n/2<\alpha \leq (3n+1)/4$.
