Veech's theorem of higher order

Abstract

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