A Kronecker-Weyl theorem for subsets of abelian groups

Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2c and a countable family E of infinite subsets of G, we construct “Baire many” monomorphisms π:G→Tc such that π(E) is dense in {y∈Tc:ny=0} whenever n∈N, E∈E, nE={0} and {x∈E:mx=g} is finite for all g∈G and m∈N∖{0} such that n=mk for some k∈N∖{1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko (2002) [22, Problem 6.5]. Applications to group actions and discrete flows on Tc, Diophantine approximation, Bohr topologies and Bohr compactifications are also provided.