  In a previous paper {GN2} an effective solution of the lattice point counting
problem in general domains in semisimple S-algebraic groups and affine
symmetric varieties was established. The method relies on the mean ergodic
theorem for the action of G on G/Gamma, and implies uniformity in counting over
families of lattice subgroups admitting a uniform spectral gap. In the present
paper we extend some methods developed in {NS} and use them to establish
several useful consequences of this property, including : Effective upper
bounds on lifting for solutions of congruences in affine homogeneous varieties,
effective upper bounds on the number of integral points on general subvarieties
of semisimple group varieties, effective lower bounds on the number of almost
prime points on symmetric varieties, and effective upper bounds on almost prime
solutions of Linnik-type congruence problems in homogeneous varieties.
