Applications of small-scale quantum ergodicity in nodal sets

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.