  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.
