Weighted geodesic restrictions of arithmetic eigenfunctions

Abstract

Let $X$ be an arithmetic hyperbolic surface, $\ddot{I}ˆ$ a Hecke-Maass form, $\ell$ a geodesic segment on $X$, and $\hat{I}¼$ a Borel measure supported on $\ell$ with dimension greater than 1/2. We obtain a power saving over the local bound of Eswarathasan and Pramanik for the $L^2$ norm of $\ddot{I}ˆ$ with respect to $\hat{I}¼$, which is a weighted generalization of Marshall's geodesic restriction bound and is proved by applying the method of arithmetic amplification. On a general 2-dimensional Riemannian manifold, we also obtain a Kakeya-Nikodym bound for the $L^2$ norm of any Laplace-Beltrami eigenfunction with respect to a Borel measure supported on a geodesic segment with dimension greater than 1/2.