Small Scale Equidistribution of Eigenfunctions on the Torus

We study the small scale distribution of the L 2 mass of eigenfunctions of the Laplacian on the flat torus Td. Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose L 2 mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions d = 3, 4 we can restrict to individual eigenspaces and show small scale equidistribution in that context. We also study irregularities of quantum equidistribution: We construct eigenfunctions whose L 2 mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension d = 4 we show the existence of eigenfunctions for which the proportion of L 2 mass in small balls blows up at certain scales.