Veech's Theorem on Higher Order Regionally Proximal Relations for Minimal Abelian Dynamical Systems

For a minimal abelian dynamical system (X, G), a point (x, y) is in the regionally proximal relation of order d, RP[d], if and only if there exists a sequence {gn} in Gd and points zε in X for each ε in {0, 1}d{0} such that the limits limn→∞(gn·ε)x = zε and limn→∞(gn·ε)−1z1 = z1-ε hold.