Inner radius of nodal domains of quantum ergodic eigenfunctions

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results [Anal. PDE 11 (2018), 855-871], which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by Nonlinearity 28 (2015), 3263-3288, Adv. Math. 290 (2016), 938-966 small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get logarithmic improvements on the inner radius of eigenfunctions on such manifolds. We also get improvements for manifolds with ergodic geodesic flows. In addition using the small scale equidistribution results of Comm. Math. Phys. 350 (2017), 279-300, one gets polynomial betterments of Comm. Partial Differential Equations 33 (2008), 1611-1621 for toral eigenfunctions in dimensions $n \geq 3$. The results work only for a full density subsequence of eigenfunctions.