Strong approximation for Zariski dense subgroups over arbitrary global fields

Consider a finitely generated Zariski dense subgroup Gamma of a connected s imple algebraic group G over a global field F. An important aspect of stron g approximation is the question of whether the closure of Gamma in the grou p of points of G with coefficients in a ring of partial adeles is open. We prove an essentially optimal result in this direction, based on the conditi on that Gamma is not discrete in that ambient group. There are no restricti ons on the characteristic of F or the type of G, and simultaneous approxima tion in finitely many algebraic groups is also studied. Classification of f inite simple groups is not used.