Tame sets, dominating maps, and complex tori

Abstract. A discrete subset of Cn is said to be tame if there is an automor-phism of Cn taking the given discrete subset to a subset of a complex line; such tame sets are known to allow interpolation by automorphisms. We give here a fairly general sufficient condition for a discrete set to be tame. In a related direction, we show that for certain discrete sets in Cn there is an injective holo-morphic map from Cn into itself whose image avoids an -neighborhood of the discrete set. Among other things, this is used to show that, given any complex n-torus and any finite set in this torus, there exist an open set containing the finite set and a locally biholomorphic map from Cn into the complement of this open set. 1