Veech's theorem of higher order

Abstract

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