Tame sets, dominating maps, and complex tori

A discrete subset of $\mathbb C^n$ is said to be tame if there is an automorphism of $\mathbb C^n$ 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 $\mathbb C^n$ there is an injective holomorphic map from $\mathbb C^n$ into itself whose image avoids an $\epsilon$-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 $\mathbb C^n$ into the complement of this open set.