Weighted geodesic restrictions of arithmetic eigenfunctions

TL;DR

The paper proves new $L^2$-restriction bounds for Hecke–Maass eigenfunctions on arithmetic hyperbolic surfaces, restricting to geodesic measures with fractal dimension $ obreak 1/2<\alpha\le 1$ and achieving a power saving in the spectral parameter $\lambda$. The approach centers on reducing the problem to a weighted geodesic restriction with an energy-controlled weight, and then applying an arithmetic amplification framework to bound the restricted norm. A precise dependence on $\alpha$ is established via the quantity $\delta(\alpha)$, yielding $\|\psi\|_{L^2(\mu)} \lesssim_ {\alpha,\varepsilon} \lambda^{1/4-\delta(\alpha)+\varepsilon}$; a Kakeya–Nikodym bound is also proved for general 2D manifolds, linking restriction to KN norms. The results advance understanding of fractal restriction phenomena for arithmetic eigenfunctions and connect to broader KN-type phenomena in spectral geometry.

Abstract

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

Theorems & Definitions (30)

• Theorem 1.1: EP22GMX24
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5