Tame sets, dominating maps, and complex tori

Abstract

A discrete subset of C n \mathbb C^n is said to be tame if there is an automorphism of C n \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 C n \mathbb C^n there is an injective holomorphic map from C n \mathbb C^n into itself whose image avoids an ϵ \epsilon -neighborhood of the discrete set. Among other things, this is used to show that, given any complex n n -torus and any finite set in this torus, there exist an open set containing the finite set and a locally biholomorphic map from C n \mathbb C^n into the complement of this open set.