  Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate
the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide
variety of subsets $\Gamma$ of $M$. The sets $\Gamma$ that we consider are
Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff
dimension.
  Our estimates are based on Frostman-type ball growth conditions for measures
supported on $\Gamma$. For large Lebesgue exponents $p$, these estimates
provide a natural generalization of $L^p$ bounds for eigenfunctions restricted
to submanifolds, previously obtained in \cite{Ho68, Ho71, Sog88, BGT07}. Under
an additional measure-theoretic assumption on $\Gamma$, the estimates are shown
to be sharp in this range. As evidence of the genericity of the sharp
estimates, we provide a large family of random, Cantor-type sets that are not
submanifolds, where the above-mentioned sharp bounds hold almost surely.
