{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ACCC3VUFAW6GE2CIBW2VBRM2JI","short_pith_number":"pith:ACCC3VUF","canonical_record":{"source":{"id":"1601.06051","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-22T15:56:22Z","cross_cats_sorted":[],"title_canon_sha256":"e4aa453044037945d732b96314962561503295d122529f2d1152d6e9f3bed49f","abstract_canon_sha256":"2cd889d16d7a7ff45ed68b70d09477c41b2ddc5688a1e558e25f3271fa14bf3e"},"schema_version":"1.0"},"canonical_sha256":"00842dd68505bc6268480db550c59a4a0501ff70e1630df64b5148d0315fba45","source":{"kind":"arxiv","id":"1601.06051","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06051","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06051v3","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06051","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"ACCC3VUFAW6G","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"ACCC3VUFAW6GE2CI","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"ACCC3VUF","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ACCC3VUFAW6GE2CIBW2VBRM2JI","target":"record","payload":{"canonical_record":{"source":{"id":"1601.06051","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-22T15:56:22Z","cross_cats_sorted":[],"title_canon_sha256":"e4aa453044037945d732b96314962561503295d122529f2d1152d6e9f3bed49f","abstract_canon_sha256":"2cd889d16d7a7ff45ed68b70d09477c41b2ddc5688a1e558e25f3271fa14bf3e"},"schema_version":"1.0"},"canonical_sha256":"00842dd68505bc6268480db550c59a4a0501ff70e1630df64b5148d0315fba45","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:54.526475Z","signature_b64":"/PAUUnGIAT+sJehNb/Q0+ELoSsvsMBnWtHuWNqBbnULpd5MkAwIPYZk0zc4W3m79sPpc3SImwm+gAvLTZo2tBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00842dd68505bc6268480db550c59a4a0501ff70e1630df64b5148d0315fba45","last_reissued_at":"2026-05-18T00:24:54.525648Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:54.525648Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.06051","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rZ80IM5vCoKdp7Fc/5SHbkZqU8UA1psifZ3oEc7rUXyqygbZattLpB0AHy6yAT6FvRAaN6QjMWEvcFgMva7hCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:39:39.747058Z"},"content_sha256":"577daddf18c0adc8edd3105d0e808f5967d9306d4c1e669d542c9fe072c169fe","schema_version":"1.0","event_id":"sha256:577daddf18c0adc8edd3105d0e808f5967d9306d4c1e669d542c9fe072c169fe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ACCC3VUFAW6GE2CIBW2VBRM2JI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative Quasiperiodicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Evelyn Sander, James A Yorke, Suddhasattwa Das, Yoshitaka Saiki","submitted_at":"2016-01-22T15:56:22Z","abstract_excerpt":"The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, $\\Sigma_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$ ergodic trajectory $(x_n)$ of a function $T$ converge to the space average $\\int f d\\mu$, where $\\mu$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the \"end\" terms when $n$ is near $0$ or $N-1$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\\infty$, we show that our weighted Birkh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06051","kind":"arxiv","version":3},"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":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JS45Pq9HQH7dIVGVxouRMt9nPDKStGx3/p1p6C7Wc37PE57tfP70yk7co9kqVkmsjAtMt/96PkWsW1MhWksgDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:39:39.747421Z"},"content_sha256":"a8c0e1d8a5ffb7396421269b09b21c2f8c345f0fe235e84258eb51c77d08ec8a","schema_version":"1.0","event_id":"sha256:a8c0e1d8a5ffb7396421269b09b21c2f8c345f0fe235e84258eb51c77d08ec8a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI/bundle.json","state_url":"https://pith.science/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI/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":"2026-06-22T23:39:39Z","links":{"resolver":"https://pith.science/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI","bundle":"https://pith.science/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI/bundle.json","state":"https://pith.science/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ACCC3VUFAW6GE2CIBW2VBRM2JI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ACCC3VUFAW6GE2CIBW2VBRM2JI","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":"2cd889d16d7a7ff45ed68b70d09477c41b2ddc5688a1e558e25f3271fa14bf3e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-22T15:56:22Z","title_canon_sha256":"e4aa453044037945d732b96314962561503295d122529f2d1152d6e9f3bed49f"},"schema_version":"1.0","source":{"id":"1601.06051","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.06051","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1601.06051v3","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06051","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"ACCC3VUFAW6G","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"ACCC3VUFAW6GE2CI","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"ACCC3VUF","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:a8c0e1d8a5ffb7396421269b09b21c2f8c345f0fe235e84258eb51c77d08ec8a","target":"graph","created_at":"2026-05-18T00:24:54Z","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":"The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, $\\Sigma_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$ ergodic trajectory $(x_n)$ of a function $T$ converge to the space average $\\int f d\\mu$, where $\\mu$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the \"end\" terms when $n$ is near $0$ or $N-1$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\\infty$, we show that our weighted Birkh","authors_text":"Evelyn Sander, James A Yorke, Suddhasattwa Das, Yoshitaka Saiki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-22T15:56:22Z","title":"Quantitative Quasiperiodicity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06051","kind":"arxiv","version":3},"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":"sha256:577daddf18c0adc8edd3105d0e808f5967d9306d4c1e669d542c9fe072c169fe","target":"record","created_at":"2026-05-18T00:24:54Z","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":"2cd889d16d7a7ff45ed68b70d09477c41b2ddc5688a1e558e25f3271fa14bf3e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-01-22T15:56:22Z","title_canon_sha256":"e4aa453044037945d732b96314962561503295d122529f2d1152d6e9f3bed49f"},"schema_version":"1.0","source":{"id":"1601.06051","kind":"arxiv","version":3}},"canonical_sha256":"00842dd68505bc6268480db550c59a4a0501ff70e1630df64b5148d0315fba45","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"00842dd68505bc6268480db550c59a4a0501ff70e1630df64b5148d0315fba45","first_computed_at":"2026-05-18T00:24:54.525648Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:54.525648Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/PAUUnGIAT+sJehNb/Q0+ELoSsvsMBnWtHuWNqBbnULpd5MkAwIPYZk0zc4W3m79sPpc3SImwm+gAvLTZo2tBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:54.526475Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.06051","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:577daddf18c0adc8edd3105d0e808f5967d9306d4c1e669d542c9fe072c169fe","sha256:a8c0e1d8a5ffb7396421269b09b21c2f8c345f0fe235e84258eb51c77d08ec8a"],"state_sha256":"9f6107fdc9c09152d001c5a996202f8f004e18c501d3a0019e6e0c1dd411ae4f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+dQB7UxmSrmT1SHMfPXgl/ScynXHqFK6SRaf7kRKzAtymXm+LANb/WhgKusAFuDQFnFjU19y48wpyN9/My3pDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:39:39.749381Z","bundle_sha256":"725e93037ac917b55bd6e7b40b65848876ab0fdeb4ac681dbcb9f1d7f9b2e368"}}