Applications of small scale quantum ergodicity in nodal sets

TL;DR

This paper explores how small scale quantum ergodicity can improve bounds on nodal sets and vanishing orders of eigenfunctions, especially on negatively curved and ergodic manifolds, using eigenfunction properties at shrinking scales.

Contribution

It demonstrates that small scale quantum ergodicity leads to improved bounds on nodal set sizes and vanishing orders of eigenfunctions, extending previous results to finer scales and specific manifold types.

Findings

Improved upper bounds on nodal set sizes based on shrinking radius r(λ).

Enhanced vanishing order estimates for eigenfunctions.

Logarithmic and o(1) improvements on negatively curved and ergodic manifolds.

Abstract

The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(\lambda) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of $r(\lambda)$. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of Han and Hezari-Rivi\`ere small scale QE holds on negatively curved manifolds at logarithmically shrinking rates, we get logarithmic improvements on such manifolds for the above measurements of eigenfunctions. We also get $o(1)$ improvements for manifolds with ergodic geodesic flows. Our results work for a full density subsequence of any given orthonormal basis of eigenfunctions.