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
then one can achieve improvements on the recent upper bounds of Logunov (2016)
and Logunov and Malinnikova (2016) on the size of nodal sets, according to a certain
power of
.
We also show that the doubling estimates and the order-of-vanishing results of
Donnelly and Fefferman (1988, 1990) can be improved. Due to results of Han (2015)
and Hezari and Rivière (2016), small-scale QE holds on negatively curved manifolds
at logarithmically shrinking rates, and thus we get logarithmic improvements on
such manifolds for the above measurements of eigenfunctions. We also get
improvements for manifolds with ergodic geodesic flows. Our results work for a full
density subsequence of any given orthonormal basis of eigenfunctions.
Keywords
eigenfunctions, nodal sets, doubling estimates, order of
vanishing, quantum ergodicity