{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:STQUOIRN77EJG2DZVPJDMFFS23","short_pith_number":"pith:STQUOIRN","canonical_record":{"source":{"id":"0904.2075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-14T09:19:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"917dd416b84f823bc7e9214dd7b12eb0ee4c9be3c563a7249ae6fb8a8b1ff5fe","abstract_canon_sha256":"8daa15d5b0c569f50006af2e351a4f251e5251a0f32f87d83ce62a44118f6587"},"schema_version":"1.0"},"canonical_sha256":"94e147222dffc8936879abd23614b2d6d962cb21b28dadfecf4fafc260c1c76b","source":{"kind":"arxiv","id":"0904.2075","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.2075","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"0904.2075v1","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.2075","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"STQUOIRN77EJ","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"pith_short_16","alias_value":"STQUOIRN77EJG2DZ","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"pith_short_8","alias_value":"STQUOIRN","created_at":"2026-07-04T15:41:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:STQUOIRN77EJG2DZVPJDMFFS23","target":"record","payload":{"canonical_record":{"source":{"id":"0904.2075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-14T09:19:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"917dd416b84f823bc7e9214dd7b12eb0ee4c9be3c563a7249ae6fb8a8b1ff5fe","abstract_canon_sha256":"8daa15d5b0c569f50006af2e351a4f251e5251a0f32f87d83ce62a44118f6587"},"schema_version":"1.0"},"canonical_sha256":"94e147222dffc8936879abd23614b2d6d962cb21b28dadfecf4fafc260c1c76b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:41:09.593283Z","signature_b64":"sDUDRx6sjfG/6G7j2RKyrMj4xMA4CGJD38uhZr0+aTO3ujyLPo59ltE6Lct53jSWi3FuC/hMiwlIMFdB1Lg+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94e147222dffc8936879abd23614b2d6d962cb21b28dadfecf4fafc260c1c76b","last_reissued_at":"2026-07-04T15:41:09.592935Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:41:09.592935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0904.2075","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":"2026-07-04T15:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JjzpXhxb5dM0chVwVE5yTKxJ9t8fFUoA0i/aUNrrQgIdd12tfMwKnEIO6CT8xjrp1J7qUprikXo8rpoW+xynBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T16:08:05.295947Z"},"content_sha256":"63ddba69346214cb5f7311b56cc7fd8237895c6e1ad87f47fb6e5f7d928a5122","schema_version":"1.0","event_id":"sha256:63ddba69346214cb5f7311b56cc7fd8237895c6e1ad87f47fb6e5f7d928a5122"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:STQUOIRN77EJG2DZVPJDMFFS23","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sum-product phenomena in F_p: a brief introduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ben Green","submitted_at":"2009-04-14T09:19:46Z","abstract_excerpt":"These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a byproduct one obtains a clean proof of the Bourgain-Katz-Tao theorem on the sum-product phenomenon in F_p. The arguments are essentially extracted from a paper of Bourgain, and I benefitted very much from being in receipt of unpublished course notes of Elon Lindenstrauss. No originality is claimed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.2075","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0904.2075/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T15:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lzvUS2H1OjlUO3Ic20vWhJUdkOLGNwta5aL1S6ojGXQLylxWRg/G4/b7+lxcnzyO7EHBVcTZt/FGd3kT3X5ABg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T16:08:05.296478Z"},"content_sha256":"a782518fdf712a797d2803b3d393755958bb554894432b36173fdb3f96f74dc9","schema_version":"1.0","event_id":"sha256:a782518fdf712a797d2803b3d393755958bb554894432b36173fdb3f96f74dc9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/STQUOIRN77EJG2DZVPJDMFFS23/bundle.json","state_url":"https://pith.science/pith/STQUOIRN77EJG2DZVPJDMFFS23/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/STQUOIRN77EJG2DZVPJDMFFS23/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":"2026-08-13T16:08:05Z","links":{"resolver":"https://pith.science/pith/STQUOIRN77EJG2DZVPJDMFFS23","bundle":"https://pith.science/pith/STQUOIRN77EJG2DZVPJDMFFS23/bundle.json","state":"https://pith.science/pith/STQUOIRN77EJG2DZVPJDMFFS23/state.json","well_known_bundle":"https://pith.science/.well-known/pith/STQUOIRN77EJG2DZVPJDMFFS23/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:STQUOIRN77EJG2DZVPJDMFFS23","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":"8daa15d5b0c569f50006af2e351a4f251e5251a0f32f87d83ce62a44118f6587","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-14T09:19:46Z","title_canon_sha256":"917dd416b84f823bc7e9214dd7b12eb0ee4c9be3c563a7249ae6fb8a8b1ff5fe"},"schema_version":"1.0","source":{"id":"0904.2075","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.2075","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"0904.2075v1","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.2075","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"STQUOIRN77EJ","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"pith_short_16","alias_value":"STQUOIRN77EJG2DZ","created_at":"2026-07-04T15:41:09Z"},{"alias_kind":"pith_short_8","alias_value":"STQUOIRN","created_at":"2026-07-04T15:41:09Z"}],"graph_snapshots":[{"event_id":"sha256:a782518fdf712a797d2803b3d393755958bb554894432b36173fdb3f96f74dc9","target":"graph","created_at":"2026-07-04T15:41:09Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0904.2075/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a byproduct one obtains a clean proof of the Bourgain-Katz-Tao theorem on the sum-product phenomenon in F_p. The arguments are essentially extracted from a paper of Bourgain, and I benefitted very much from being in receipt of unpublished course notes of Elon Lindenstrauss. No originality is claimed.","authors_text":"Ben Green","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-14T09:19:46Z","title":"Sum-product phenomena in F_p: a brief introduction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.2075","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":"sha256:63ddba69346214cb5f7311b56cc7fd8237895c6e1ad87f47fb6e5f7d928a5122","target":"record","created_at":"2026-07-04T15:41:09Z","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":"8daa15d5b0c569f50006af2e351a4f251e5251a0f32f87d83ce62a44118f6587","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-04-14T09:19:46Z","title_canon_sha256":"917dd416b84f823bc7e9214dd7b12eb0ee4c9be3c563a7249ae6fb8a8b1ff5fe"},"schema_version":"1.0","source":{"id":"0904.2075","kind":"arxiv","version":1}},"canonical_sha256":"94e147222dffc8936879abd23614b2d6d962cb21b28dadfecf4fafc260c1c76b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94e147222dffc8936879abd23614b2d6d962cb21b28dadfecf4fafc260c1c76b","first_computed_at":"2026-07-04T15:41:09.592935Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:41:09.592935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sDUDRx6sjfG/6G7j2RKyrMj4xMA4CGJD38uhZr0+aTO3ujyLPo59ltE6Lct53jSWi3FuC/hMiwlIMFdB1Lg+DA==","signature_status":"signed_v1","signed_at":"2026-07-04T15:41:09.593283Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.2075","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:63ddba69346214cb5f7311b56cc7fd8237895c6e1ad87f47fb6e5f7d928a5122","sha256:a782518fdf712a797d2803b3d393755958bb554894432b36173fdb3f96f74dc9"],"state_sha256":"a1ef0a2d112fbcb30a2344bbfb0a673323d6622552cb45556798b2eed4cb1b07"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wmgN1QSJII/lb2UV+UdFHrD6WUEgmJf+aBveMBxKB+i93idUfaHiKIfPURuvAaY7av3IblImWVz3A3kf70PJBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T16:08:05.300760Z","bundle_sha256":"ebac2c46fa25696dd7d72137af42a319802801d574381e7702d856ff901dabb1"}}