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

By S. Shao, Xiangdong Ye. 2010

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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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53a7b42a31eabefd0f66fa1ca27f72df512a8856

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Nil Bohr-sets and almost automorphy of higher order (2014, methodology): “Recently, Shao and Ye [48] proved that the above conclusion holds for general minimal systems.”
On automorphism groups of low complexity subshifts (2015, background): “Here, we consider the maximal nilfactor ([20],[35]) of a minimal subshift to find a class of examples with arbitrarily big polynomial complexity whose group of automorphisms is virtually Z.”