{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BROUFS2RZNCEEJXCOENQBNDSJ2","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":"bdae87c8c3307b30ff920451593efab92c29bf9f2f6fc8d2f833900d28201dd2","cross_cats_sorted":["math.CA","math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2016-04-28T15:34:11Z","title_canon_sha256":"844d18cf8a869c516a5fa72ef4b5e9fa22b33bd02611f9e0c2c7a7034357c274"},"schema_version":"1.0","source":{"id":"1604.08469","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.08469","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"arxiv_version","alias_value":"1604.08469v4","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08469","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"pith_short_12","alias_value":"BROUFS2RZNCE","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BROUFS2RZNCEEJXC","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BROUFS2R","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:867eb5523f10439fe475731499905bee3ea1fa0a2c0de904c40c672893e0b246","target":"graph","created_at":"2026-05-18T00:51:03Z","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":"We use an estimate of Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) on the number of solutions of certain equations involving products and differences of sets in prime finite fields to give an explicit upper bound on trilinear exponential sums which improves the previous bound of Bourgain and Garaev (2009). We also obtain explicit bounds for quadrilinear exponential sums.","authors_text":"Giorgis Petridis, Igor E. Shparlinski","cross_cats":["math.CA","math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2016-04-28T15:34:11Z","title":"Bounds on trilinear and quadrilinear exponential sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08469","kind":"arxiv","version":4},"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":"sha256:5e59c6e44ee3853c8108ec3750ba3881d97d268b232094f5eee84f79b33b1664","target":"record","created_at":"2026-05-18T00:51:03Z","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":"bdae87c8c3307b30ff920451593efab92c29bf9f2f6fc8d2f833900d28201dd2","cross_cats_sorted":["math.CA","math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2016-04-28T15:34:11Z","title_canon_sha256":"844d18cf8a869c516a5fa72ef4b5e9fa22b33bd02611f9e0c2c7a7034357c274"},"schema_version":"1.0","source":{"id":"1604.08469","kind":"arxiv","version":4}},"canonical_sha256":"0c5d42cb51cb444226e2711b00b4724ebe9eaab65b6ad1bbdcb8eb594afb9f46","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c5d42cb51cb444226e2711b00b4724ebe9eaab65b6ad1bbdcb8eb594afb9f46","first_computed_at":"2026-05-18T00:51:03.256088Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:03.256088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OZdhBdn0msHA/89fI2WDxNPZsBzO7vQHhiwom6xVWVe7dg1hT6xwU7JDa/LHRxRwCg0YStUx9/Fh3iGQHQWkCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:03.256585Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.08469","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e59c6e44ee3853c8108ec3750ba3881d97d268b232094f5eee84f79b33b1664","sha256:867eb5523f10439fe475731499905bee3ea1fa0a2c0de904c40c672893e0b246"],"state_sha256":"cdca0443ed531e5678ee4db6868c494c9fbad0e3f68dec2c661d9aeb6516e6c2"}