Abstract
We study the small scale distribution of the L 2-mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300, 2017) showed the existence of a density one subsequence whose L 2-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L 2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.
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Communicated by J. Marklof
Dedicated to the memory of Javier Cilleruelo
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Granville, A., Wigman, I. Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions. Commun. Math. Phys. 355, 767–802 (2017). https://doi.org/10.1007/s00220-017-2953-3
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DOI: https://doi.org/10.1007/s00220-017-2953-3