{"paper":{"title":"Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.NT"],"primary_cat":"math.CA","authors_text":"Ben Krause","submitted_at":"2014-02-08T00:31:30Z","abstract_excerpt":"Let $L^2(X,\\Sigma,\\mu,\\tau)$ be a measure-preserving system, with $\\tau$ a $\\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \\[ M_N(f):= \\frac{1}{N}\\sum_{n \\leq N} \\tau^{P(n)} f \\] converge pointwise for $f \\in L^2(X)$. We do so by proving that, for $r>2$, the $r$-variation, $\\mathcal{V}^r(M_N(f))$, extends to a bounded operator on $L^2$. We also prove that our result is sharp, in that $\\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1803","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"}