{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XHRFO4OZRJHM6PWNKHYZBDUCLO","short_pith_number":"pith:XHRFO4OZ","canonical_record":{"source":{"id":"1007.5249","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-07-29T15:32:54Z","cross_cats_sorted":["cs.LO","math.PR"],"title_canon_sha256":"452759d5e6c911177876fa0c291d6edea5da0c997441745a0934328d6f6ee5ff","abstract_canon_sha256":"e51d8a1a5756b6b5a323373d2918c0afc427cb4935c168f8613b1748b7396e44"},"schema_version":"1.0"},"canonical_sha256":"b9e25771d98a4ecf3ecd51f1908e825badb00c063fb75839d513188db13f0b78","source":{"kind":"arxiv","id":"1007.5249","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.5249","created_at":"2026-05-18T04:16:12Z"},{"alias_kind":"arxiv_version","alias_value":"1007.5249v2","created_at":"2026-05-18T04:16:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.5249","created_at":"2026-05-18T04:16:12Z"},{"alias_kind":"pith_short_12","alias_value":"XHRFO4OZRJHM","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XHRFO4OZRJHM6PWN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XHRFO4OZ","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XHRFO4OZRJHM6PWNKHYZBDUCLO","target":"record","payload":{"canonical_record":{"source":{"id":"1007.5249","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-07-29T15:32:54Z","cross_cats_sorted":["cs.LO","math.PR"],"title_canon_sha256":"452759d5e6c911177876fa0c291d6edea5da0c997441745a0934328d6f6ee5ff","abstract_canon_sha256":"e51d8a1a5756b6b5a323373d2918c0afc427cb4935c168f8613b1748b7396e44"},"schema_version":"1.0"},"canonical_sha256":"b9e25771d98a4ecf3ecd51f1908e825badb00c063fb75839d513188db13f0b78","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:12.166765Z","signature_b64":"Ew+ZuxAWWQWPUqrLWjUMG0NJsarVRzxUu7xqIinZNyAEQWhkRT3JE8iL1150WZtCtRETxxcip48HMJOtxBK3CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9e25771d98a4ecf3ecd51f1908e825badb00c063fb75839d513188db13f0b78","last_reissued_at":"2026-05-18T04:16:12.166356Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:12.166356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.5249","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xwd2WTyTSyTfTbdjdRc9zLkD3Yo/7Z8M0ZT51u2Feq0NCEhmeX4fAqyMeKYJnJ8wSc3qOzWtji6kbZHJcyu+BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:10:40.796802Z"},"content_sha256":"05770dda3ed8396d5e4a861725e0ae6b8d6763a550579aae53e8792f32e5c281","schema_version":"1.0","event_id":"sha256:05770dda3ed8396d5e4a861725e0ae6b8d6763a550579aae53e8792f32e5c281"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XHRFO4OZRJHM6PWNKHYZBDUCLO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A constructive version of Birkhoff's ergodic theorem for Martin-L\\\"of random points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO","math.PR"],"primary_cat":"math.DS","authors_text":"Adam Day, Alexander Shen, Ilya Mezhirov, Laurent Bienvenu, Mathieu Hoyrup","submitted_at":"2010-07-29T15:32:54Z","abstract_excerpt":"A theorem of Ku\\v{c}era states that given a Martin-L\\\"of random infinite binary sequence {\\omega} and an effectively open set A of measure less than 1, some tail of {\\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.5249","kind":"arxiv","version":2},"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":"2026-05-18T04:16:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7hoa3hYvKyfGUUKZKn+KK+djirl1JnZHoEo4uoqAFAQEcM98vmYLOY05jrTDpj+k3L8WbXttm34bdl2Y8s3sAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:10:40.797507Z"},"content_sha256":"7c63ad113e1e2ecb7aa2e9e81c9a2cd485920448df2aa32043ae7abf9db45479","schema_version":"1.0","event_id":"sha256:7c63ad113e1e2ecb7aa2e9e81c9a2cd485920448df2aa32043ae7abf9db45479"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO/bundle.json","state_url":"https://pith.science/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO/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":"2026-05-25T22:10:40Z","links":{"resolver":"https://pith.science/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO","bundle":"https://pith.science/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO/bundle.json","state":"https://pith.science/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XHRFO4OZRJHM6PWNKHYZBDUCLO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XHRFO4OZRJHM6PWNKHYZBDUCLO","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":"e51d8a1a5756b6b5a323373d2918c0afc427cb4935c168f8613b1748b7396e44","cross_cats_sorted":["cs.LO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-07-29T15:32:54Z","title_canon_sha256":"452759d5e6c911177876fa0c291d6edea5da0c997441745a0934328d6f6ee5ff"},"schema_version":"1.0","source":{"id":"1007.5249","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.5249","created_at":"2026-05-18T04:16:12Z"},{"alias_kind":"arxiv_version","alias_value":"1007.5249v2","created_at":"2026-05-18T04:16:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.5249","created_at":"2026-05-18T04:16:12Z"},{"alias_kind":"pith_short_12","alias_value":"XHRFO4OZRJHM","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XHRFO4OZRJHM6PWN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XHRFO4OZ","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:7c63ad113e1e2ecb7aa2e9e81c9a2cd485920448df2aa32043ae7abf9db45479","target":"graph","created_at":"2026-05-18T04:16:12Z","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":"A theorem of Ku\\v{c}era states that given a Martin-L\\\"of random infinite binary sequence {\\omega} and an effectively open set A of measure less than 1, some tail of {\\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.","authors_text":"Adam Day, Alexander Shen, Ilya Mezhirov, Laurent Bienvenu, Mathieu Hoyrup","cross_cats":["cs.LO","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-07-29T15:32:54Z","title":"A constructive version of Birkhoff's ergodic theorem for Martin-L\\\"of random points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.5249","kind":"arxiv","version":2},"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":"sha256:05770dda3ed8396d5e4a861725e0ae6b8d6763a550579aae53e8792f32e5c281","target":"record","created_at":"2026-05-18T04:16:12Z","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":"e51d8a1a5756b6b5a323373d2918c0afc427cb4935c168f8613b1748b7396e44","cross_cats_sorted":["cs.LO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-07-29T15:32:54Z","title_canon_sha256":"452759d5e6c911177876fa0c291d6edea5da0c997441745a0934328d6f6ee5ff"},"schema_version":"1.0","source":{"id":"1007.5249","kind":"arxiv","version":2}},"canonical_sha256":"b9e25771d98a4ecf3ecd51f1908e825badb00c063fb75839d513188db13f0b78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9e25771d98a4ecf3ecd51f1908e825badb00c063fb75839d513188db13f0b78","first_computed_at":"2026-05-18T04:16:12.166356Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:12.166356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ew+ZuxAWWQWPUqrLWjUMG0NJsarVRzxUu7xqIinZNyAEQWhkRT3JE8iL1150WZtCtRETxxcip48HMJOtxBK3CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:12.166765Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.5249","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05770dda3ed8396d5e4a861725e0ae6b8d6763a550579aae53e8792f32e5c281","sha256:7c63ad113e1e2ecb7aa2e9e81c9a2cd485920448df2aa32043ae7abf9db45479"],"state_sha256":"a05804d1da49a81546749d39a82948fb7e7b1f30715740f2081c018c751b54b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/smP5kSyjzevMp5GYWG3IM1dwVTItI4Aya4fFGArQ2CVvPLNE4rl/XMirup5hfGb/C7Zn5YCe+XcEznjvqnPCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T22:10:40.800859Z","bundle_sha256":"23f55a36e8dbcd08510fd48846f0174fe4d246003f5cd804361628054d9fcdf5"}}