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Martin Barlow (personal communication) asked the following question, which arose from the work of a student (Zichun Ye) on interface models. Question: If $f(x) \\ge 0$ is integrable, and $N(x) = \\min \\{n: A_kf(x) \\le 2 \\int f \\text{for all} k \\ge n\\}$, is it the case that $N(x)$ is also integr"},"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":"1609.05560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-09-18T22:10:44Z","cross_cats_sorted":[],"title_canon_sha256":"4f4f7bbab71e951726ac56092fd9d1c78a41f7350893fb0f34a66502ed7af917","abstract_canon_sha256":"f2b4d1c5fc3224768105d1d1a94488a9e5fbca93fff1518e7b4b3f4411a044c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:26.047230Z","signature_b64":"XiWzSIq77SyHz7bCbg96NxMZYFFuIbBnDY1RiTAHd++63rFEn6oHFi5TIrGKWuYJB2k/6/kzYxIwAOb0ujfPBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1477e403ca4ae3c3810c603cbab2c1e4aeda39f313bfad573d65321482e9f1de","last_reissued_at":"2026-05-18T01:04:26.046567Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:26.046567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Slowly decaying averages and fat towers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"James T. Campbell, M\\'at\\'e Wierdl","submitted_at":"2016-09-18T22:10:44Z","abstract_excerpt":"Let $(X,\\Sigma,m,\\tau)$ be an ergodic system, that is, $(X, \\Sigma, m)$ is a probability space and $\\tau: X \\to X$ is an invertible ergodic $m$-preserving transformation. For a function $f:X\\to\\mathbb R$, let $A_Nf$ denote the $N$th ergodic average, $A_Nf(x)=\\frac{1}{N}\\cdot (f(x)+\\dots+\\tau^ {N-1}f(x))$. Martin Barlow (personal communication) asked the following question, which arose from the work of a student (Zichun Ye) on interface models. Question: If $f(x) \\ge 0$ is integrable, and $N(x) = \\min \\{n: A_kf(x) \\le 2 \\int f \\text{for all} k \\ge n\\}$, is it the case that $N(x)$ is also integr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05560","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":"1609.05560","created_at":"2026-05-18T01:04:26.046691+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.05560v1","created_at":"2026-05-18T01:04:26.046691+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05560","created_at":"2026-05-18T01:04:26.046691+00:00"},{"alias_kind":"pith_short_12","alias_value":"CR36IA6KJLR4","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"CR36IA6KJLR4HAIM","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"CR36IA6K","created_at":"2026-05-18T12:30:09.641336+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/CR36IA6KJLR4HAIMMA6LVMWB4S","json":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S.json","graph_json":"https://pith.science/api/pith-number/CR36IA6KJLR4HAIMMA6LVMWB4S/graph.json","events_json":"https://pith.science/api/pith-number/CR36IA6KJLR4HAIMMA6LVMWB4S/events.json","paper":"https://pith.science/paper/CR36IA6K"},"agent_actions":{"view_html":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S","download_json":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S.json","view_paper":"https://pith.science/paper/CR36IA6K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.05560&json=true","fetch_graph":"https://pith.science/api/pith-number/CR36IA6KJLR4HAIMMA6LVMWB4S/graph.json","fetch_events":"https://pith.science/api/pith-number/CR36IA6KJLR4HAIMMA6LVMWB4S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/action/storage_attestation","attest_author":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/action/author_attestation","sign_citation":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/action/citation_signature","submit_replication":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/action/replication_record"}},"created_at":"2026-05-18T01:04:26.046691+00:00","updated_at":"2026-05-18T01:04:26.046691+00:00"}