Higher order almost automorphy, recurrence sets and the regionally proximal relation

In this paper, $d$-step almost automorphic systems are studied for $d\in\N$, which are the generalization of the classical almost automorphic ones. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$ is $d$-step almost automorphic can be characterized via various subsets of $\Z$ including the dual sets of $d$-step Poincar\'e and Birkhoff recurrence sets, and Nil$_d$ Bohr$_0$-sets by considering $N(x,V)=\{n\in\Z: T^nx\in V\}$, where $V$ is an arbitrary neighborhood of $x$. Moreover, it turns out that the condition $(x,y)\in X\times X$ is regionally proximal of order $d$ can also be characterized via various subsets of $\Z$ including $d$-step Poincar\'e and Birkhoff recurrence sets, $SG_d$ sets, the dual sets of Nil$_d$ Bohr$_0$-sets, and others by considering $N(x,U)=\{n\in\Z: T^nx\in U\}$, where $U$ is an arbitrary neighborhood of $y$.