{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:OSZEGLJT4U5QZBXKBB4YT4V6RK","short_pith_number":"pith:OSZEGLJT","schema_version":"1.0","canonical_sha256":"74b2432d33e53b0c86ea087989f2be8abd4904401a41e2674ebb43d3d74d0b44","source":{"kind":"arxiv","id":"1402.1803","version":1},"attestation_state":"computed","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 $L^2$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1402.1803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-08T00:31:30Z","cross_cats_sorted":["math.DS","math.NT"],"title_canon_sha256":"00a32b6a40441337b87d01f9575084a5a83680732f24f8e2d1b81832097f112d","abstract_canon_sha256":"5261c52cb3b047f3ab04aadb504bbe003a8415d715e03bcfe64321ed08709215"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:33.513156Z","signature_b64":"nhKR74XtODHhoedoPCCc0EteXoPTanM+zKS0eUciK8krf7nUwTM9GiL12JgiOo0Zwmel38Lv/IK+ERWU8iV3CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74b2432d33e53b0c86ea087989f2be8abd4904401a41e2674ebb43d3d74d0b44","last_reissued_at":"2026-05-18T02:59:33.512462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:33.512462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1402.1803","created_at":"2026-05-18T02:59:33.512579+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.1803v1","created_at":"2026-05-18T02:59:33.512579+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.1803","created_at":"2026-05-18T02:59:33.512579+00:00"},{"alias_kind":"pith_short_12","alias_value":"OSZEGLJT4U5Q","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"OSZEGLJT4U5QZBXK","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"OSZEGLJT","created_at":"2026-05-18T12:28:43.426989+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK","json":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK.json","graph_json":"https://pith.science/api/pith-number/OSZEGLJT4U5QZBXKBB4YT4V6RK/graph.json","events_json":"https://pith.science/api/pith-number/OSZEGLJT4U5QZBXKBB4YT4V6RK/events.json","paper":"https://pith.science/paper/OSZEGLJT"},"agent_actions":{"view_html":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK","download_json":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK.json","view_paper":"https://pith.science/paper/OSZEGLJT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.1803&json=true","fetch_graph":"https://pith.science/api/pith-number/OSZEGLJT4U5QZBXKBB4YT4V6RK/graph.json","fetch_events":"https://pith.science/api/pith-number/OSZEGLJT4U5QZBXKBB4YT4V6RK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK/action/storage_attestation","attest_author":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK/action/author_attestation","sign_citation":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK/action/citation_signature","submit_replication":"https://pith.science/pith/OSZEGLJT4U5QZBXKBB4YT4V6RK/action/replication_record"}},"created_at":"2026-05-18T02:59:33.512579+00:00","updated_at":"2026-05-18T02:59:33.512579+00:00"}