AI 中文总结
该论文解答了Xu和Ye关于可数离散群的伯努利转移中远端点稠密性的问题，引入点根基概念，证明相关群类性质并构造出答案存在差异的2步幂零群。
AI 中文摘要
我们解答了Xu和Ye（《群作用下与所有极小系统不交》，待发表于《以色列数学杂志》，arXiv:2212.07830）提出的关于可数离散群G的伯努利转移2^G中远端点稠密性的问题。针对一个相关但更强的概念——几乎自守点，我们通过证明对应的群类与极大殆周期群类一致，解答了类似问题。这些刻画使我们能够构造出对Xu-Ye问题答案存在差异的2步幂零群。为寻找Xu-Ye问题的内蕴解答，我们引入了可数离散群的点根基概念，并证明必要条件是点根基为平凡的。最后，我们考虑了一些相关问题，证明了所有满足远端点集合在2^G中稠密的可数群G构成的类在有限指标扩张下是闭的，且满足常序列是唯一的远端（几乎自守）点的可数群G构成的类与极小殆周期群类一致。
英文摘要
We give an answer to a question of Xu and Ye (Disjointness with all minimal systems under group actions, to appear in Israel J. Math., arxiv:2212.07830) on the denseness of distal points in the Bernoulli shift $2^G$ for a countable discrete group $G$. For a related but stronger notion of almost automorphic points, we answer the similar question by showing that the corresponding collection of the groups coincides with maximal almost periodic ones. These characterizations allow us to construct 2-step nilpotent groups for which the answers to the Xu-Ye question differ. In search for an intrinsic answer to the Xu-Ye question, we introduce a notion of point-distal radical for a countable discrete group and show that a necessary condition is for the point-distal radical to be trivial. Finally, we consider some related questions, and show that the collection of all countable groups $G$ for which the set of distal points is dense in $2^G$ is closed under finite-index extension, and that the collection of countable groups $G$ for which the constant sequences are the only distal (almost automorphic) points coincides with the minimally almost periodic ones.
