Some questions in Diophantine approximation: real and p-adics

The Weak approximation theorem describes the closure of $G(Q)$ inside G(Qₚ) as well as inside $G(R)$ for G an algebraic group over Q; the closure is always an open normal subgroup with finite abelian quotient, and is well understood in a certain sense even if precise results are not always available (such as for tori!). In this paper, for a finitely generated subgroup L ⊂ G(Q) we consider the topological closure of L inside G(Qₚ) and $G(R)$. The paper is written mostly for G a torus or an abelian variety, but eventually considers a variant of the question for G a semisimple group. The paper is written with the wishful thinking that when dealing with questions on topological closure of algebraic points in an algebraic group defined over a number field, the simplest answers hold, a well-known principle known as ``Occum's razor''.