Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov’s potential density

Abstract:

For an uncountable cardinal $\tau$ and a subset $S$ of an abelian group $G$, the following conditions are equivalent:

[(i)] $|\{ns:s\in S\}|\ge \tau$ for all integers $n\ge 1$;

[(ii)] there exists a group homomorphism $\pi :G\to \mathbb {T}^{2^\tau }$ such that $\pi (S)$ is dense in $\mathbb {T}^{2^\tau }$.

Moreover, if $|G|\le 2^{2^\tau }$, then the following item can be added to this list:

[(iii)] there exists an isomorphism $\pi :G\to G’$ between $G$ and a subgroup $G’$ of $\mathbb {T}^{2^\tau }$ such that $\pi (S)$ is dense in $\mathbb {T}^{2^\tau }$.

We prove that the following conditions are equivalent for an uncountable subset $S$ of an abelian group $G$ that is either (almost) torsion-free or divisible:

[(a)] $S$ is $\mathscr {T}$-dense in $G$ for some Hausdorff group topology $\mathscr {T}$ on $G$;

[(b)] $S$ is $\mathscr {T}$-dense in some precompact Hausdorff group topology $\mathscr {T}$ on $G$;

[(c)] $|\{ns:s\in S\}|\ge \min \left \{\tau :|G|\le 2^{2^\tau }\right \}$ for every integer $n\ge 1$.

This partially resolves a question of Markov going back to 1946.