Automorphic density estimates and optimal Diophantine exponents

The present paper is devoted to establishing an optimal approximation exponent for the action of an irreducible uniform lattice subgroup of a product group on its proper factors. Previously optimal approximation exponents for lattice actions on homogeneous spaces were established under the assumption that the restriction of the automorphic representation to the stability group is suitably tempered. However, for irreducible lattices in semisimple algebraic groups, either this property does not hold or it amounts to an instance of the Ramanujan-Petersson-Selberg conjecture. Sarnak's Density Hypothesis and its variants bounding the multiplicities of irreducible representations occurring in the decomposition of the automorphic representation can be viewed as a weakening of the temperedness property. A refined form of this hypothesis has recently been established for uniform irreducible arithmetic congruence lattices arising from quaternion algebras. We employ this result in order to establish - unconditionally - an optimal approximation exponent for the actions of these lattices on the associated symmetric spaces. We also give a general spectral criterion for the optimality of the approximation exponent for irreducible uniform lattices in a product of arbitrary Gelfand pairs. Our methods involve utilizing the multiplicity bounds in the pre-trace formula, establishing refined estimates of the spherical transforms, and carrying out an elaborate spectral analysis that bounds the Hilbert-Schmidt norms of carefully balanced geometric convolution operators.