A Kronecker-Weyl theorem for subsets of abelian groups

TL;DR

This paper extends the Kronecker-Weyl theorem to subsets of abelian groups, constructing dense embeddings into tori, and applies this to algebraic and topological problems, including Markov's problem and group actions.

Contribution

It introduces a new construction of dense monomorphisms from abelian groups into tori, advancing the understanding of potentially dense subsets and solving longstanding problems.

Findings

Constructed monomorphisms dense in specified subsets

Provided an algebraic description of potentially dense subsets

Solved a problem of Markov from 1944 and addressed Tkachenko and Yaschenko's question

Abstract

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 F of infinite subsets of G, we construct "Baire many" monomorphisms p: G --> T^c such that p(E) is dense in {y in T^c : ny=0} whenever n in N, E in F, nE={0} and {x in E: mx=g} is finite for all g in G and m such that n=mk for some k in 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. Applications to group actions and discrete flows on T^c, diophantine approximation, Bohr topologies and Bohr compactifications are also provided.