Congruence properties of Zariski-dense subgroups

This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of K-rational points of G. Then what should it mean for F to be a 'large ' subgroup? We might require F to be a lattice in G, to be arithmetic, to contain many elements of a specifi