In [Gorodnik and Nevo, Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176] 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/Γ, 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 [Nevo and Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361–402] and use them to establish several useful consequences of this property, including:
(1) effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;
(2) effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;
(3) effective lower bounds on the number of almost prime points on symmetric varieties;
(4) effective upper bounds on almost prime solutions of congruences in homogeneous varieties.
