Veech's theorem of higher order

Abstract: For an abelian group $G$, $\vec{g}=(g_1,\ldots,g_d)\in G^d$ and $\epsilon=(\epsilon(1),\ldots,\epsilon(d))\in {0,1}^d$, 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, $(x,y)\in \mathbf{RP}^{[d]}$ 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 $\epsilon\in {0,1}^d\backslash{ \vec{0}}$, [ \lim_{n\to\infty}(\vec{g}n\cdot\epsilon)x= z\epsilon\quad \mathrm{and} \quad \lim_{n\to\infty}(\vec{g}n\cdot\epsilon)^{-1}z{\vec{1}}=z_{\vec{1}-\epsilon}, ] where $\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$.