{"paper":{"title":"Nil Bohr$_0$-sets, Poincar\\'e recurrence and generalized polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","submitted_at":"2011-09-16T14:49:50Z","abstract_excerpt":"The problem which can be viewed as the higher order version of an old question concerning Bohr sets is investigated: for any $d\\in \\N$ does the collection of $\\{n\\in \\Z: S\\cap (S-n)\\cap...\\cap (S-dn)\\neq \\emptyset\\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets?\n  In this paper it is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem could be answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\\supset \\{n\\in \\Z: S\\cap (S-n)\\cap...\\cap (S-dn)\\neq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3636","kind":"arxiv","version":1},"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"}