{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:RAUDTQS3NU7B2ZEVG5PT2B4IWL","short_pith_number":"pith:RAUDTQS3","canonical_record":{"source":{"id":"1905.00291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-01T12:56:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ac1d58c67070fc4759530e7361e611c20f37a0dab827458d26292e4299853fed","abstract_canon_sha256":"fd979642cb44750aa98b03c8365d94f92006aa1f40d194ba84de4ff16dbc19cb"},"schema_version":"1.0"},"canonical_sha256":"882839c25b6d3e1d6495375f3d0788b2d5c38478470d96d97fce0cff3a308ca6","source":{"kind":"arxiv","id":"1905.00291","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.00291","created_at":"2026-05-17T23:47:13Z"},{"alias_kind":"arxiv_version","alias_value":"1905.00291v1","created_at":"2026-05-17T23:47:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.00291","created_at":"2026-05-17T23:47:13Z"},{"alias_kind":"pith_short_12","alias_value":"RAUDTQS3NU7B","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"RAUDTQS3NU7B2ZEV","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"RAUDTQS3","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:RAUDTQS3NU7B2ZEVG5PT2B4IWL","target":"record","payload":{"canonical_record":{"source":{"id":"1905.00291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-01T12:56:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ac1d58c67070fc4759530e7361e611c20f37a0dab827458d26292e4299853fed","abstract_canon_sha256":"fd979642cb44750aa98b03c8365d94f92006aa1f40d194ba84de4ff16dbc19cb"},"schema_version":"1.0"},"canonical_sha256":"882839c25b6d3e1d6495375f3d0788b2d5c38478470d96d97fce0cff3a308ca6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:13.247655Z","signature_b64":"OGmwsBz1lFbRm/F7lOF/ZwiiHx1gPZVVlUbLeCEXTfb+OVYijYHlKPCvEvvrbWdkO+eyFgmVZ27SVXd/IWb6Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"882839c25b6d3e1d6495375f3d0788b2d5c38478470d96d97fce0cff3a308ca6","last_reissued_at":"2026-05-17T23:47:13.247075Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:13.247075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1905.00291","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":"2026-05-17T23:47:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"62W1pO7do++DafXS/i3eN5ccwWzEX0aa5k5IQqIS3CDAn7Eetr0Ny2K9WMQWmJYeq3jCKM4uBLrTc6MSADZgDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T10:03:47.612616Z"},"content_sha256":"ff106364f764c5952ea62db52bf121f2e44ff56a4e33a4e5f99159be67ee9c2a","schema_version":"1.0","event_id":"sha256:ff106364f764c5952ea62db52bf121f2e44ff56a4e33a4e5f99159be67ee9c2a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:RAUDTQS3NU7B2ZEVG5PT2B4IWL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular hyperbolas and bilinear forms of Kloosterman sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ilya D. Shkredov","submitted_at":"2019-05-01T12:56:28Z","abstract_excerpt":"In this paper we study incidences for hyperbolas in $\\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of Kloosterman sums."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.00291","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":"2026-05-17T23:47:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0H2un9lIYeDvn/+GbroGQDqYQNK9HVt9/GbPAPR868MWzbJf8qWOQITWiS8AiaZogzPyEaGa0pVLhEQu0DW5Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T10:03:47.613290Z"},"content_sha256":"f719f1b1584d3359164bd4e2b425ad73b6fd24882ac2a467cc9f9eb3d041cca7","schema_version":"1.0","event_id":"sha256:f719f1b1584d3359164bd4e2b425ad73b6fd24882ac2a467cc9f9eb3d041cca7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL/bundle.json","state_url":"https://pith.science/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL/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":"2026-05-26T10:03:47Z","links":{"resolver":"https://pith.science/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL","bundle":"https://pith.science/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL/bundle.json","state":"https://pith.science/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RAUDTQS3NU7B2ZEVG5PT2B4IWL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:RAUDTQS3NU7B2ZEVG5PT2B4IWL","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":"fd979642cb44750aa98b03c8365d94f92006aa1f40d194ba84de4ff16dbc19cb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-01T12:56:28Z","title_canon_sha256":"ac1d58c67070fc4759530e7361e611c20f37a0dab827458d26292e4299853fed"},"schema_version":"1.0","source":{"id":"1905.00291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.00291","created_at":"2026-05-17T23:47:13Z"},{"alias_kind":"arxiv_version","alias_value":"1905.00291v1","created_at":"2026-05-17T23:47:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.00291","created_at":"2026-05-17T23:47:13Z"},{"alias_kind":"pith_short_12","alias_value":"RAUDTQS3NU7B","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"RAUDTQS3NU7B2ZEV","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"RAUDTQS3","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:f719f1b1584d3359164bd4e2b425ad73b6fd24882ac2a467cc9f9eb3d041cca7","target":"graph","created_at":"2026-05-17T23:47:13Z","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":"In this paper we study incidences for hyperbolas in $\\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of Kloosterman sums.","authors_text":"Ilya D. Shkredov","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-01T12:56:28Z","title":"Modular hyperbolas and bilinear forms of Kloosterman sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.00291","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:ff106364f764c5952ea62db52bf121f2e44ff56a4e33a4e5f99159be67ee9c2a","target":"record","created_at":"2026-05-17T23:47:13Z","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":"fd979642cb44750aa98b03c8365d94f92006aa1f40d194ba84de4ff16dbc19cb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-01T12:56:28Z","title_canon_sha256":"ac1d58c67070fc4759530e7361e611c20f37a0dab827458d26292e4299853fed"},"schema_version":"1.0","source":{"id":"1905.00291","kind":"arxiv","version":1}},"canonical_sha256":"882839c25b6d3e1d6495375f3d0788b2d5c38478470d96d97fce0cff3a308ca6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"882839c25b6d3e1d6495375f3d0788b2d5c38478470d96d97fce0cff3a308ca6","first_computed_at":"2026-05-17T23:47:13.247075Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:47:13.247075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OGmwsBz1lFbRm/F7lOF/ZwiiHx1gPZVVlUbLeCEXTfb+OVYijYHlKPCvEvvrbWdkO+eyFgmVZ27SVXd/IWb6Ag==","signature_status":"signed_v1","signed_at":"2026-05-17T23:47:13.247655Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.00291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ff106364f764c5952ea62db52bf121f2e44ff56a4e33a4e5f99159be67ee9c2a","sha256:f719f1b1584d3359164bd4e2b425ad73b6fd24882ac2a467cc9f9eb3d041cca7"],"state_sha256":"1efab32ef7b15934784a5b08350fa671bb01227145324d627fedda64d136e641"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yur67fDG/O7F0ZK2u9QFHKc6pD8oz0708BK3HIG658wKF8oiilR/tnQQbZqgQBvSmduUQ4mCmpCtymCi57sHBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T10:03:47.617023Z","bundle_sha256":"aa2aa98833d3c5c8007a9bc74f3d63c8575dcd2590fa95866b4e2fc1d2b92e2d"}}