{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CR36IA6KJLR4HAIMMA6LVMWB4S","short_pith_number":"pith:CR36IA6K","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"},"canonical_sha256":"1477e403ca4ae3c3810c603cbab2c1e4aeda39f313bfad573d65321482e9f1de","source":{"kind":"arxiv","id":"1609.05560","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.05560","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"arxiv_version","alias_value":"1609.05560v1","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05560","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"pith_short_12","alias_value":"CR36IA6KJLR4","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CR36IA6KJLR4HAIM","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CR36IA6K","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CR36IA6KJLR4HAIMMA6LVMWB4S","target":"record","payload":{"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"},"canonical_sha256":"1477e403ca4ae3c3810c603cbab2c1e4aeda39f313bfad573d65321482e9f1de","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"},"source_kind":"arxiv","source_id":"1609.05560","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:04:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X7yT4p26nM0SZ7G0Ucmk/IuKpP5mS16nHpZz2CRShCngf8i608IvUScISWnykgW8eUyg/8vgCe9HMq3yD7ZZAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:49:39.842866Z"},"content_sha256":"d248457dc1ab29bbdd344ab9c3fb47a8726041d194408e045f77e52a14f9c4af","schema_version":"1.0","event_id":"sha256:d248457dc1ab29bbdd344ab9c3fb47a8726041d194408e045f77e52a14f9c4af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CR36IA6KJLR4HAIMMA6LVMWB4S","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:04:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PKfy0MjPobY3yuKNAl+kW37Gb11vY/ZOAcS/R4sPGzYCpMPD/tVCypxPsTgHM713s3vxw8yqwEiIrSCKbHxRBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:49:39.843214Z"},"content_sha256":"f884760c3c2c3b108233c56d9dc6b6eed2f619eb7ca3a1fc35bcfc674f1c4901","schema_version":"1.0","event_id":"sha256:f884760c3c2c3b108233c56d9dc6b6eed2f619eb7ca3a1fc35bcfc674f1c4901"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/bundle.json","state_url":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T03:49:39Z","links":{"resolver":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S","bundle":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/bundle.json","state":"https://pith.science/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CR36IA6KJLR4HAIMMA6LVMWB4S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CR36IA6KJLR4HAIMMA6LVMWB4S","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f2b4d1c5fc3224768105d1d1a94488a9e5fbca93fff1518e7b4b3f4411a044c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-09-18T22:10:44Z","title_canon_sha256":"4f4f7bbab71e951726ac56092fd9d1c78a41f7350893fb0f34a66502ed7af917"},"schema_version":"1.0","source":{"id":"1609.05560","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.05560","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"arxiv_version","alias_value":"1609.05560v1","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05560","created_at":"2026-05-18T01:04:26Z"},{"alias_kind":"pith_short_12","alias_value":"CR36IA6KJLR4","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CR36IA6KJLR4HAIM","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CR36IA6K","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:f884760c3c2c3b108233c56d9dc6b6eed2f619eb7ca3a1fc35bcfc674f1c4901","target":"graph","created_at":"2026-05-18T01:04:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"James T. Campbell, M\\'at\\'e Wierdl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-09-18T22:10:44Z","title":"Slowly decaying averages and fat towers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05560","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d248457dc1ab29bbdd344ab9c3fb47a8726041d194408e045f77e52a14f9c4af","target":"record","created_at":"2026-05-18T01:04:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f2b4d1c5fc3224768105d1d1a94488a9e5fbca93fff1518e7b4b3f4411a044c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-09-18T22:10:44Z","title_canon_sha256":"4f4f7bbab71e951726ac56092fd9d1c78a41f7350893fb0f34a66502ed7af917"},"schema_version":"1.0","source":{"id":"1609.05560","kind":"arxiv","version":1}},"canonical_sha256":"1477e403ca4ae3c3810c603cbab2c1e4aeda39f313bfad573d65321482e9f1de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1477e403ca4ae3c3810c603cbab2c1e4aeda39f313bfad573d65321482e9f1de","first_computed_at":"2026-05-18T01:04:26.046567Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:26.046567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XiWzSIq77SyHz7bCbg96NxMZYFFuIbBnDY1RiTAHd++63rFEn6oHFi5TIrGKWuYJB2k/6/kzYxIwAOb0ujfPBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:26.047230Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.05560","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d248457dc1ab29bbdd344ab9c3fb47a8726041d194408e045f77e52a14f9c4af","sha256:f884760c3c2c3b108233c56d9dc6b6eed2f619eb7ca3a1fc35bcfc674f1c4901"],"state_sha256":"af1835ce0b32b1abbb524d44e488faf2d4c507ab78cdd96b9a611da6a762e38b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZGdBUiACgW6Yew4p9O2TDAc8V3eC2/bBgSEDK+TvR8D2fW+VxgCsmEMT9IPGaUsShOgEIjCz1luAJK3mlvllAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T03:49:39.845428Z","bundle_sha256":"40da9bfcbcb0aaeddde73b2f1b9726ae839a4ed900b3956263aa1736aec855f1"}}