{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PR3OQKHHA64TIFYPIL42RBW3GW","short_pith_number":"pith:PR3OQKHH","canonical_record":{"source":{"id":"1807.10998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-29T04:56:43Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"ac3fd211d460fb4efcba4ce959e8cbbd180d6948e3176fddb494b301cd514dd1","abstract_canon_sha256":"6449474aabefc60bd0d235a3efc2ec20f4f1fa6592509c5706758352648fbd6c"},"schema_version":"1.0"},"canonical_sha256":"7c76e828e707b934170f42f9a886db35909a36901a4dad4976c7203bb29f76bd","source":{"kind":"arxiv","id":"1807.10998","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.10998","created_at":"2026-05-18T00:09:33Z"},{"alias_kind":"arxiv_version","alias_value":"1807.10998v1","created_at":"2026-05-18T00:09:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10998","created_at":"2026-05-18T00:09:33Z"},{"alias_kind":"pith_short_12","alias_value":"PR3OQKHHA64T","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PR3OQKHHA64TIFYP","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PR3OQKHH","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PR3OQKHHA64TIFYPIL42RBW3GW","target":"record","payload":{"canonical_record":{"source":{"id":"1807.10998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-29T04:56:43Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"ac3fd211d460fb4efcba4ce959e8cbbd180d6948e3176fddb494b301cd514dd1","abstract_canon_sha256":"6449474aabefc60bd0d235a3efc2ec20f4f1fa6592509c5706758352648fbd6c"},"schema_version":"1.0"},"canonical_sha256":"7c76e828e707b934170f42f9a886db35909a36901a4dad4976c7203bb29f76bd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:33.213336Z","signature_b64":"05MpbUVS0vgHipl5EcF7pAAskjGhUp6b0AqxneGSt3cDlWBvVChL/cllkOfAq5IIQxov/CBp2Kdd8/t9prrWCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c76e828e707b934170f42f9a886db35909a36901a4dad4976c7203bb29f76bd","last_reissued_at":"2026-05-18T00:09:33.212759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:33.212759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.10998","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":"2026-05-18T00:09:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DNU0Y8uw3KknOJkWbD9lm2rSH0A0xMwDIv6zcwpGdvqSVuFyxyuaDMI1k+M8bLNlkw2o3430B9osIoURrWa5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:19:48.207014Z"},"content_sha256":"46a87d8c9f02deec502bb65fd20cf7bd3e56be3de94421e22a5c5c461fbbb850","schema_version":"1.0","event_id":"sha256:46a87d8c9f02deec502bb65fd20cf7bd3e56be3de94421e22a5c5c461fbbb850"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PR3OQKHHA64TIFYPIL42RBW3GW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A new sum-product estimate in prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ali Mohammadi, Bryce Kerr, Changhao Chen","submitted_at":"2018-07-29T04:56:43Z","abstract_excerpt":"In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if $A\\subseteq \\mathbb{F}_p$ satisfies $|A|\\le p^{64/117}$ then $$ \\max\\{|A\\pm A|, |AA|\\} \\gtrsim |A|^{39/32}. $$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^+(P)$ to a point-plane incidence bound of Rudnev rather than a point line incidence bound of Stevens and de "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10998","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":"2026-05-18T00:09:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ENERlATu9/KMjacp9zEmpQ/yUGECu4AGmgJeL5xhY+5fzXLo88a3aJjPmJT+63/h+Add/EJ2cO/XuSMMtx5XCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T18:19:48.207760Z"},"content_sha256":"389706dfa4717b56d89a85d109f27bde09b998474e0bbabab7eec50b1df05315","schema_version":"1.0","event_id":"sha256:389706dfa4717b56d89a85d109f27bde09b998474e0bbabab7eec50b1df05315"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PR3OQKHHA64TIFYPIL42RBW3GW/bundle.json","state_url":"https://pith.science/pith/PR3OQKHHA64TIFYPIL42RBW3GW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PR3OQKHHA64TIFYPIL42RBW3GW/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":"2026-05-29T18:19:48Z","links":{"resolver":"https://pith.science/pith/PR3OQKHHA64TIFYPIL42RBW3GW","bundle":"https://pith.science/pith/PR3OQKHHA64TIFYPIL42RBW3GW/bundle.json","state":"https://pith.science/pith/PR3OQKHHA64TIFYPIL42RBW3GW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PR3OQKHHA64TIFYPIL42RBW3GW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PR3OQKHHA64TIFYPIL42RBW3GW","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":"6449474aabefc60bd0d235a3efc2ec20f4f1fa6592509c5706758352648fbd6c","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-29T04:56:43Z","title_canon_sha256":"ac3fd211d460fb4efcba4ce959e8cbbd180d6948e3176fddb494b301cd514dd1"},"schema_version":"1.0","source":{"id":"1807.10998","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.10998","created_at":"2026-05-18T00:09:33Z"},{"alias_kind":"arxiv_version","alias_value":"1807.10998v1","created_at":"2026-05-18T00:09:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10998","created_at":"2026-05-18T00:09:33Z"},{"alias_kind":"pith_short_12","alias_value":"PR3OQKHHA64T","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PR3OQKHHA64TIFYP","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PR3OQKHH","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:389706dfa4717b56d89a85d109f27bde09b998474e0bbabab7eec50b1df05315","target":"graph","created_at":"2026-05-18T00:09:33Z","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 obtain a new sum-product estimate in prime fields. In particular, we show that if $A\\subseteq \\mathbb{F}_p$ satisfies $|A|\\le p^{64/117}$ then $$ \\max\\{|A\\pm A|, |AA|\\} \\gtrsim |A|^{39/32}. $$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^+(P)$ to a point-plane incidence bound of Rudnev rather than a point line incidence bound of Stevens and de ","authors_text":"Ali Mohammadi, Bryce Kerr, Changhao Chen","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-29T04:56:43Z","title":"A new sum-product estimate in prime fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10998","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":"sha256:46a87d8c9f02deec502bb65fd20cf7bd3e56be3de94421e22a5c5c461fbbb850","target":"record","created_at":"2026-05-18T00:09:33Z","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":"6449474aabefc60bd0d235a3efc2ec20f4f1fa6592509c5706758352648fbd6c","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-29T04:56:43Z","title_canon_sha256":"ac3fd211d460fb4efcba4ce959e8cbbd180d6948e3176fddb494b301cd514dd1"},"schema_version":"1.0","source":{"id":"1807.10998","kind":"arxiv","version":1}},"canonical_sha256":"7c76e828e707b934170f42f9a886db35909a36901a4dad4976c7203bb29f76bd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c76e828e707b934170f42f9a886db35909a36901a4dad4976c7203bb29f76bd","first_computed_at":"2026-05-18T00:09:33.212759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:33.212759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"05MpbUVS0vgHipl5EcF7pAAskjGhUp6b0AqxneGSt3cDlWBvVChL/cllkOfAq5IIQxov/CBp2Kdd8/t9prrWCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:33.213336Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.10998","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46a87d8c9f02deec502bb65fd20cf7bd3e56be3de94421e22a5c5c461fbbb850","sha256:389706dfa4717b56d89a85d109f27bde09b998474e0bbabab7eec50b1df05315"],"state_sha256":"774af689c22ac557aa301c7e0a09a3dc5b0f4b4c94fa7d24e5862f6dde642c9f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GsFkSwPJ36akr5vtos8tSzH4wwt1T0Q8JLhoNVWCfpXbLbDwWCxVN5qTe2lvYhmqUf0YBAToefJTR9eJG7SiDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T18:19:48.211709Z","bundle_sha256":"e7137b8a44076997b212053ff38bdd390b13f847f12e332c9c3ffb2a35b6a551"}}