Planck-scale number of nodal domains for toral eigenfunctions

Abstract

We study the number of nodal domains in balls shrinking slightly above the Planck scale for "generic" toral eigenfunctions. We prove that, up to the natural scaling, the nodal domains count obeys the same asymptotic law as the global number of nodal domains. The proof, on one hand, uses new arithmetic information to refine Bourgain's de-randomisation technique at Planck scale. And on the other hand, it requires a Planck scale version of Yau's conjecture which we believe to be of independent interest.