On the denseness of distal points

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.