  We study the small scale distribution of the $L^2$-mass of eigenfunctions of
the Laplacian on the the two-dimensional flat torus. Given an orthonormal basis
of eigenfunctions, Lester and Rudnick 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.
