Arithmetic Homogeneous Spaces

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can be considered pointwise versions (on a general manifold) of Hardy's lower bounds for the error term in the Gauss circle problem. Our results develop and extend the unpublished thesis of A. Karnaukh. This is joint work with I. Polterovich (Univ. of Montreal)