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 2 c and a countable family E of infinite subsets of G, we construct “Baire many” monomorphisms π : G → T c such that π ( E ) is dense in { y ∈ T c : n y = 0 } whenever n ∈ N , E ∈ E , n E = { 0 } and { x ∈ E : m x = g } is finite for all g ∈ G and m ∈ N ∖ { 0 } such that n = m k 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 T c , Diophantine approximation, Bohr topologies and Bohr compactifications are also provided.