  We study the fractal pointwise convergence for the equation
$i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and
non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with
$s>(n-\alpha+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^\alpha$-a.e,
where $\mathcal{H}^\alpha$ is the $\alpha$-dimensional Hausdorff measure. We
improve upon this result depending on the dispersive strength of $P$. On the
other hand, for a family of polynomials $P$ and given $\alpha$, we exploit a
Talbot-like effect to construct initial data whose solutions $u$ diverge in
sets of Hausdorff dimension $\alpha$. To compute the dimension of the sets of
divergence, we adopt the Mass Transference Principle from Diophantine
approximation. We also construct counterexamples for quadratic symbols like the
saddle to show that our positive results are sometimes best possible.
