THE BLOCK APPROXIMATION THEOREM

The block approximation theorem is an extensive general- ization of both the well known weak approximation theorem from valu- ation theory and the density property of global fields in their henseliza- tions. It guarantees the existence of rational points of smooth affine varieties that solve approximation problems of local-global type (see e.g. [HJP07]). The theorem holds for pseudo real closed fields, by [FHV94]. In this paper we prove the block approximation for pseudo-F- closed fields K, where F is an ´etale compact family of valuations of K with bounded residue fields (Theorem 4.1). This includes in particular the case of pseudo p-adically closed fields and generalizations of these like the ones considered in [HJP05].