closure of units of number fields in the finite idele topology

Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units. Further let us denote by $\mathbb A _{K,f}^\times$ the ideles of $K$ together with its usual idele topology and by $\widehat{\mathcal O}_K^\times$ its standard compact subset of integral, finite, ideles.

What can be said about the closures $\overline {\mathcal O _K ^\times}$ and $\overline{\mathcal O _{K,+}^\times}$ taken in $\widehat{\mathcal O}_K^\times$?

Is it possible to describe the closures explicitly?

For me the most interesting case are real quadratic fields for the beginning. If I remember correctly for imaginary quadratic fields the units should be discrete.

Doesn't strong approximation for $\mathbb{A}$ assert that they lie dense? –  plusepsilon.de Mar 17 '12 at 20:00