Veech's theorem of higher order

Abstract

For an abelian group $G$, $\vec{g}=(g_1,\dots ,g_d)\in G^d$ and $\hat{I}\mu =(\hat{I}\mu (1),\dots ,\hat{I}\mu (d))\in \{0,1{\}}^d$, let $\vec{g}\cdot \hat{I}\mu ={\prod }_{i=1}^d{g}_i^{\hat{I}\mu (i)}$. 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 $z_{\hat{I}}\mu \in X,\hat{I}\mu \in \{0,1{\}}^d$ with $z_{\vec{0}}=y$ such that for every $\hat{I}\mu \in \{0,1{\}}^d\backslash \{\vec{0}\}$, \[ \lim_{n\to\infty}(\vec{g}_n\cdotε)x= z_ε\quad \mathrm{and} \quad \lim_{n\to\infty}(\vec{g}_n\cdotε)^{-1}z_{\vec{1}}=z_{\vec{1}-ε}, \] where ${\mathbf{RP}}^{[d]}$ is the regionally proximal relation of order $d$.