Veech's theorem of higher order

For an abelian group $G$, $\vec{g}=(g_1,\dots ,g_d)\in G^d$ and $\epsilon =(\epsilon (1),\dots ,\epsilon (d))\in \{0,1{\}}^d$, let $\vec{g}\cdot \epsilon ={\prod }_{i=1}^d{g}_i^{\epsilon (i)}$. In this paper, it is shown that for a minimal system $(X,G)$ with $G$ being abelian, $(x,y)\in {\mathbf{R}\mathbf{P}}^{[d]}$ if and only if there exists a sequence $\{{\vec{g}}_n{\}}_{n\in \mathbb{N}}\subseteq G^d$ and points $z_{\epsilon }\in X,\epsilon \in \{0,1{\}}^d$ with $z_{\vec{0}}=y$ such that for every $\epsilon \in \{0,1{\}}^d\mathrm{\backslash }\{\vec{0}\}$, $$\underset{n\to \mathrm{\infty }}{\lim }({\vec{g}}_n\cdot \epsilon )x=z_{\epsilon }\mathrm{a}\mathrm{n}\mathrm{d}\underset{n\to \mathrm{\infty }}{\lim }({\vec{g}}_n\cdot \epsilon {)}^{-1}{z}_{\vec{1}}=z_{\vec{1}-\epsilon },$$ where ${\mathbf{R}\mathbf{P}}^{[d]}$ is the regionally proximal relation of order $d$.