{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:5I2W7TTQA6RXDTQEB7YLLNWI7E","short_pith_number":"pith:5I2W7TTQ","canonical_record":{"source":{"id":"0709.3562","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-09-22T04:32:19Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"44cd354167a6db37918a62661be8a4f56fa09136917a197df79d1c579f6297fb","abstract_canon_sha256":"8dfc2d0e5ee320a628f20bff53384ca4a6dfdeb762e01d7f37229164fb10bd26"},"schema_version":"1.0"},"canonical_sha256":"ea356fce7007a371ce040ff0b5b6c8f924d17e9209b6587a1ae18ae1aecd0b9c","source":{"kind":"arxiv","id":"0709.3562","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.3562","created_at":"2026-05-18T01:35:21Z"},{"alias_kind":"arxiv_version","alias_value":"0709.3562v6","created_at":"2026-05-18T01:35:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.3562","created_at":"2026-05-18T01:35:21Z"},{"alias_kind":"pith_short_12","alias_value":"5I2W7TTQA6RX","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"5I2W7TTQA6RXDTQE","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"5I2W7TTQ","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:5I2W7TTQA6RXDTQEB7YLLNWI7E","target":"record","payload":{"canonical_record":{"source":{"id":"0709.3562","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-09-22T04:32:19Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"44cd354167a6db37918a62661be8a4f56fa09136917a197df79d1c579f6297fb","abstract_canon_sha256":"8dfc2d0e5ee320a628f20bff53384ca4a6dfdeb762e01d7f37229164fb10bd26"},"schema_version":"1.0"},"canonical_sha256":"ea356fce7007a371ce040ff0b5b6c8f924d17e9209b6587a1ae18ae1aecd0b9c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:21.025650Z","signature_b64":"c2vyJHPo4zhJcPn+On9xXOEPLPEH9nM1KSeXmA4LWGr1VWBwoh8bKqht4xHBHj7wY+hWChYyGsAyExQB09poDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea356fce7007a371ce040ff0b5b6c8f924d17e9209b6587a1ae18ae1aecd0b9c","last_reissued_at":"2026-05-18T01:35:21.024773Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:21.024773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0709.3562","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:35:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/Dhi3H9XEgbWLYbW8jz3Igbri+1x6U/y0oLDM1bUouhTvCGGO7w/SvJ976xyfc6hrtehy+MILrwuTcFqmR1hAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:04:38.669535Z"},"content_sha256":"9f6f88a8a6960ba4ebcc415ae6a7a67ad76a28169ecf2a3f69be9d1f843bff3b","schema_version":"1.0","event_id":"sha256:9f6f88a8a6960ba4ebcc415ae6a7a67ad76a28169ecf2a3f69be9d1f843bff3b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:5I2W7TTQA6RXDTQEB7YLLNWI7E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The quantitative behaviour of polynomial orbits on nilmanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Ben Green, Terence Tao","submitted_at":"2007-09-22T04:32:19Z","abstract_excerpt":"A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\\ldots)$ on a nilmanifold $G/\\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \\epsilon g'\\gamma$, where $\\epsilon(n)$ is \"smooth\", $\\gamma(n)$ is periodic and \"rational\", and $(g'(a),g'(a+d),\\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.3562","kind":"arxiv","version":6},"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":"2026-05-18T01:35:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U+pKf7dUiN2LMSDVW3Qzv6zg/Imjds6Y/8Q2oWKegDFVMkleOowPF9blOpvkmi0LyZbUepEz2Kp9C7t0vCy5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:04:38.670222Z"},"content_sha256":"9b38f06e8ca1a65ed2d32ff67e76afc8562a715fffd3adf8d2a07f339c6fa972","schema_version":"1.0","event_id":"sha256:9b38f06e8ca1a65ed2d32ff67e76afc8562a715fffd3adf8d2a07f339c6fa972"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E/bundle.json","state_url":"https://pith.science/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E/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":"2026-06-04T23:04:38Z","links":{"resolver":"https://pith.science/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E","bundle":"https://pith.science/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E/bundle.json","state":"https://pith.science/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5I2W7TTQA6RXDTQEB7YLLNWI7E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:5I2W7TTQA6RXDTQEB7YLLNWI7E","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":"8dfc2d0e5ee320a628f20bff53384ca4a6dfdeb762e01d7f37229164fb10bd26","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-09-22T04:32:19Z","title_canon_sha256":"44cd354167a6db37918a62661be8a4f56fa09136917a197df79d1c579f6297fb"},"schema_version":"1.0","source":{"id":"0709.3562","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.3562","created_at":"2026-05-18T01:35:21Z"},{"alias_kind":"arxiv_version","alias_value":"0709.3562v6","created_at":"2026-05-18T01:35:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.3562","created_at":"2026-05-18T01:35:21Z"},{"alias_kind":"pith_short_12","alias_value":"5I2W7TTQA6RX","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"5I2W7TTQA6RXDTQE","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"5I2W7TTQ","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:9b38f06e8ca1a65ed2d32ff67e76afc8562a715fffd3adf8d2a07f339c6fa972","target":"graph","created_at":"2026-05-18T01:35:21Z","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 Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\\ldots)$ on a nilmanifold $G/\\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \\epsilon g'\\gamma$, where $\\epsilon(n)$ is \"smooth\", $\\gamma(n)$ is periodic and \"rational\", and $(g'(a),g'(a+d),\\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a s","authors_text":"Ben Green, Terence Tao","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-09-22T04:32:19Z","title":"The quantitative behaviour of polynomial orbits on nilmanifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.3562","kind":"arxiv","version":6},"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":"sha256:9f6f88a8a6960ba4ebcc415ae6a7a67ad76a28169ecf2a3f69be9d1f843bff3b","target":"record","created_at":"2026-05-18T01:35:21Z","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":"8dfc2d0e5ee320a628f20bff53384ca4a6dfdeb762e01d7f37229164fb10bd26","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-09-22T04:32:19Z","title_canon_sha256":"44cd354167a6db37918a62661be8a4f56fa09136917a197df79d1c579f6297fb"},"schema_version":"1.0","source":{"id":"0709.3562","kind":"arxiv","version":6}},"canonical_sha256":"ea356fce7007a371ce040ff0b5b6c8f924d17e9209b6587a1ae18ae1aecd0b9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ea356fce7007a371ce040ff0b5b6c8f924d17e9209b6587a1ae18ae1aecd0b9c","first_computed_at":"2026-05-18T01:35:21.024773Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:21.024773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c2vyJHPo4zhJcPn+On9xXOEPLPEH9nM1KSeXmA4LWGr1VWBwoh8bKqht4xHBHj7wY+hWChYyGsAyExQB09poDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:21.025650Z","signed_message":"canonical_sha256_bytes"},"source_id":"0709.3562","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f6f88a8a6960ba4ebcc415ae6a7a67ad76a28169ecf2a3f69be9d1f843bff3b","sha256:9b38f06e8ca1a65ed2d32ff67e76afc8562a715fffd3adf8d2a07f339c6fa972"],"state_sha256":"ea641f89072629644262f4e256e02e6331f1eaf082c3ec00c74880b87efba3fd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jbwOuEskyggq22fvkbRPbECeqVBCpwiOEvsnSw5cZQ1MTRoKtgxNXkRb88nyOsX+VEPrYlDIIEU49b6SNrMBBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T23:04:38.673922Z","bundle_sha256":"6b33ea30fd783ea61b278c7dd500aceb8d844c6fc5d425cb647ea601806e269e"}}