{
  "id": "1007.0189",
  "version": "v2",
  "published": "2010-07-01T14:57:29.000Z",
  "updated": "2010-11-06T14:53:26.000Z",
  "title": "Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence",
  "authors": [
    "Song Shao",
    "Xiangdong Ye"
  ],
  "comment": "34 pages, 2 figures, the final version for submission",
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-06T14:53:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37A99"
    ],
    "keywords": [
      "regionally proximal relation",
      "minimal systems",
      "combinatorial consequence",
      "equivalence",
      "positive upper banach density subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.0189S"
    }
  }
}