Characterization of the tori via density of the solution set of linear equations (Q2738748)

Let \(G\) be a compact metrizable Abelian group with an invariant metric \(d\). The authors introduce the property \((\mathcal E)\): for each \(a\in G\), there exists a sequence \(\delta_{n}\) converging to \(0\) such that \(S_{n}(a):=\{x\in G\mid nx=a\}\) is \(\delta_{n}\)-dense in \(G\) (that is, every element \(y\in G\) has at most distance \(\delta_{n}\) from the set \(S_{n}(a)\)). Clearly, every torus \({\mathbb T}^{m}\) has property \((\mathcal E)\). In the paper under review, it is shown that a compact Abelian group \(G\) is a torus (of finite dimension) if, and only if, property \((\mathcal E)\) holds for every one-dimensional closed connected subgroup of \(G\). However, the infinite power \({\mathbb T}^\omega\) satisfies \((\mathcal E)\), and there do exist \(2\)-dimensional compact connected Abelian groups satisfying \((\mathcal E)\) that are not isomorphic to \({\mathbb T}^2\). NEWLINENEWLINENEWLINEIt is also shown that, for compact metrizable Abelian groups, property \((\mathcal E)\) is equivalent to any one of the following. NEWLINENEWLINENEWLINE\((*)\): There exists a sequence \(\varepsilon_{n}\) converging to \(0\) from above such that \(S_{n}(0)\) is \(\varepsilon_{n}\)-dense in \(G\) for each \(n\). NEWLINENEWLINENEWLINE\((\mathcal D)\): For each prime \(p\) and for each infinite set \(\pi\) of primes, the \(p\)-torsion subgroup \(t_{p}(G)\) and the \(\pi\)-socle \(soc_\pi(G)\) of \(G\) are dense in \(G\). NEWLINENEWLINENEWLINEFinally, property \((\mathcal E)\) is generalized to a sequence of conditions \((\mathcal E_m)\), involving solutions for equations of the form \(k_{1}x_{1}+\cdots k_{m}x_{m}=a\). It is shown that tori of finite dimension satisfy these conditions. The impact of this result for the construction of countably compact group topologies is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].