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

TL;DR

This paper extends classical theorems on abelian groups by establishing equivalences between density conditions, homomorphism existence, and set cardinalities, partially resolving a longstanding question of Markov from 1946.

Contribution

It generalizes Hewitt-Marczewski-Pondiczery type theorems to uncountable cardinals and provides new characterizations of potential density in abelian groups, addressing Markov's question.

Findings

Equivalence between density conditions and homomorphism existence for uncountable subsets.

Characterization of potential density in terms of set cardinalities and topologies.

Partial resolution of Markov's 1946 question on 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 T^{2^\tau} such that \pi(S) is dense in 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 T^{2^\tau} such that \pi(S) is dense in 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 T-dense in G for some Hausdorff group topology T on G; (b) S is T-dense in some precompact Hausdorff group topology T on G; (c) |{ns:s\in S}|\ge \min{\tau:|G|\le 2^{2^\tau}} for every integer n\ge 1. This partially resolves a question of Markov going back to 1946.