{"paper":{"title":"Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Xiangdong Ye","submitted_at":"2010-07-01T14:57:29Z","abstract_excerpt":"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.\n  A combinatorial consequence is that if $S$ is a dynamically syndetic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0189","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}