{"paper":{"title":"The quantitative behaviour of polynomial orbits on nilmanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Ben Green, Terence Tao","submitted_at":"2007-09-22T04:32:19Z","abstract_excerpt":"A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\\ldots)$ on a nilmanifold $G/\\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \\epsilon g'\\gamma$, where $\\epsilon(n)$ is \"smooth\", $\\gamma(n)$ is periodic and \"rational\", and $(g'(a),g'(a+d),\\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.3562","kind":"arxiv","version":6},"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"}