Weighted geodesic restrictions of arithmetic eigenfunctions

TL;DR

This paper establishes improved bounds for the $L^2$ norms of arithmetic eigenfunctions restricted to geodesic segments, using arithmetic amplification, and extends results to general Riemannian surfaces with measures of dimension greater than 1/2.

Contribution

It introduces a power-saving bound for weighted geodesic restrictions of arithmetic eigenfunctions and generalizes Kakeya-Nikodym bounds to broader Riemannian settings.

Findings

Power saving over local bounds for eigenfunction restrictions

Extension of bounds to measures with dimension > 1/2 on general surfaces

Application of arithmetic amplification method

Abstract

Let $X$ be an arithmetic hyperbolic surface, $\psi$ a Hecke-Maass form, $\ell$ a geodesic segment on $X$, and $\mu$ 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 $\psi$ with respect to $\mu$, 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.