Independence and almost automorphy of high order

In this paper, it is shown that for a minimal system $(X,T)$ and $d,kin mathbb{N}$, if $(x,x_i)$ is regionally proximal of order $d$ for $1leq ileq k$, then $(x,x_1,ldots,x_k)$ is $(k+1)$-regionally proximal of order $d$. Meanwhile, we introduce the notion of $mathrm{IN}^{[d]}$-pair: for a dynamical system $(X,T)$ and $din mathbb{N}$, a pair $(x_0,x_1)in Xtimes X$ is called an $mathrm{IN}^{[d]}$-pair if for any $kin mathbb{N}$ and any neighborhoods $U_0 ,U_1 $ of $x_0$ and $x_1$ respectively, there exist integers $p_j^{(i)},1leq ileq k,$ $1leq jleq d$ such that $$ bigcup_{i=1}^k{ p_1^{(i)}epsilon(1)+ldots+p_d^{(i)} epsilon(d):epsilon(j)in {0,1},1leq jleq d}backslash {0}subset mathrm{Ind}(U_0,U_1), $$ where $mathrm{Ind}(U_0,U_1)$ denotes the collection of all independence sets for $(U_0,U_1)$. It turns out that for a minimal system, if it dose not contain any nontrivial $mathrm{IN}^{[d]}$-pair, then it is an almost one-to-one extension of its maximal factor of order $d$.