Characterizations of groups generated by Kronecker sets

In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus T = R/Z by subsets of Z. Here we consider new types of subgroups: let K ⊆ T be a Kronecker set (a compact set on which every continuous function f : K → T can be uniformly approximated by characters of T ), and G the group generated by K. We prove (Theorem 1) that G can be characterized by a subset of Z (instead of a subset of Z). If K is finite, Theorem 1 implies our earlier result in [B-S]. We also prove (Theorem 2) that if K is uncountable, then G cannot be characterized by a subset of Z (or an integer sequence) in the sense of [B-D-S].