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

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

(i)

$\vert\{ns:s\in S\}\vert\ge \tau$ 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 $\vert G\vert\le 2^{2^\tau}$, 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)

$\vert\{ns:s\in S\}\vert\ge \min\left\{\tau:\vert G\vert\le 2^{2^\tau}\right\}$ for every integer $n\ge 1$.

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