Restriction of Laplace-Beltrami eigenfunctions to arbitrary sets on manifolds

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 $\mathrm{\Gamma }$ of $M$. The sets $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$. 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 $\mathrm{\Gamma }$, 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.