Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence

Abstract: By proving the minimality of face transformations acting on the diagonal points and searching the points allowed in the minimal sets, it is shown that the regionally proximal relation of order $d$, $\RP^{[d]}$, is an equivalence relation for minimal systems. Moreover, the lifting of $\RP^{[d]}$ between two minimal systems is obtained, which implies that the factor induced by $\RP^{[d]}$ is the maximal $d$-step nilfactor. The above results extend the same conclusions proved by Host, Kra and Maass for minimal distal systems. A combinatorial consequence is that if $S$ is a dynamically syndetic subset of $\Z$, then for each $d\ge 1$, $${(n_1,...,n_d)\in \Z^d: n_1\ep_1+... +n_d\ep_d\in S, \ep_i\in {0,1}, 1\le i\le d}$$ is syndetic. In some sense this is the topological correspondence of the result obtained by Host and Kra for positive upper Banach density subsets using ergodic methods.