关于阿贝尔群子集的Kronecker-Weyl定理
A Kronecker-Weyl theorem for subsets of abelian groups
摘要：
设N为非负整数的集合，T为循环群，c为连续统的基数.给定一个大小至多为2 ^c的阿贝尔群G和G的无限子集的可数族F，我们构造“Baire many”单态p：G --> T^c使得p（E）在{y in T^c：ny=0}中稠密，只要n在N中，E在F中，nE={0}且{x in E：mx=g}对G中的所有g和m是有限的，使得对N中的某个k，n= n-{1}。我们应用这一结果得到一个代数描述可数潜在的稠密子集的阿贝尔群，从而取得了重大进展的解决问题的马尔可夫回到1944年。我们的结果的一个特殊情况下产生一个肯定的答案Tkachenko和Yaschenko的问题。还提供了T^c、丢番图逼近、玻尔拓扑和玻尔紧化在群作用和离散流中的应用。
英文摘要：
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.
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