{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:6Y2QQD5BJVMYCVYASCFP6U5NWX","short_pith_number":"pith:6Y2QQD5B","canonical_record":{"source":{"id":"math/0610893","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2006-10-29T14:01:46Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"406900962a4141ff50e28f2a6119a8d673adc63a4b563f97349cc4189b083068","abstract_canon_sha256":"a22f32041e7d5c612ac0579ba851ec974f6bb5b3493e0060b613a8338a67dc4f"},"schema_version":"1.0"},"canonical_sha256":"f635080fa14d59815700908aff53adb5cc8172f964d03febdae54fe4f1997972","source":{"kind":"arxiv","id":"math/0610893","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610893","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610893v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610893","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"6Y2QQD5BJVMY","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6Y2QQD5BJVMYCVYA","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6Y2QQD5B","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:6Y2QQD5BJVMYCVYASCFP6U5NWX","target":"record","payload":{"canonical_record":{"source":{"id":"math/0610893","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2006-10-29T14:01:46Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"406900962a4141ff50e28f2a6119a8d673adc63a4b563f97349cc4189b083068","abstract_canon_sha256":"a22f32041e7d5c612ac0579ba851ec974f6bb5b3493e0060b613a8338a67dc4f"},"schema_version":"1.0"},"canonical_sha256":"f635080fa14d59815700908aff53adb5cc8172f964d03febdae54fe4f1997972","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:22.534260Z","signature_b64":"ryOvFtVpNY3XCUlYoDdqFLMQwOAc3riVLrWLA9Bgweizcd5n4PCorhoBPE432P6HBxjiBS3Y7aXNebqBbf2qCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f635080fa14d59815700908aff53adb5cc8172f964d03febdae54fe4f1997972","last_reissued_at":"2026-05-18T01:05:22.533511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:22.533511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0610893","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":"2026-05-18T01:05:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7LC1tmCCttgqyiY1QUWgcFYVcQFtZxiktGsdOCRymNNm382/o+a4PkqRjnzcQsFU9W6RAUuqwwxVNA5Y0jiRDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T00:26:54.467512Z"},"content_sha256":"bbc1fcd7bf84de5ef442c98fddac082a3fd82d6865f8a422998b1b0743901055","schema_version":"1.0","event_id":"sha256:bbc1fcd7bf84de5ef442c98fddac082a3fd82d6865f8a422998b1b0743901055"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:6Y2QQD5BJVMYCVYASCFP6U5NWX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Snevily's conjecture and restricted sumsets","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2006-10-29T14:01:46Z","abstract_excerpt":"Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every positive integer m\\leq (k-1)/(n-1) there are more than (k-1)n-(m+1)n(n-1)/2 sets {a_1,...,a_n} such that a_1\\in A_1,..., a_n\\in A_n, and both a_i\\not=a_j and ma_i+b_i\\not=ma_j+b_j (or both ma_i\\not=ma_j and a_i+b_i\\not=a_j+b_j) for all 1\\leq i<j\\leq n.\n This extends a recent result of Dasgupta, K\\'arolyi, Serra and\n Szegedy on Snevily's conjecture. Actually stron"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610893","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":"2026-05-18T01:05:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9qSKkXfE2B057EhvrFMtpRzwAbJ5KoblHDS55RkChzkfzDXeZtxVW5ucicoiYyhgrZZVDZGr8D1MfiDFdzdEAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T00:26:54.467873Z"},"content_sha256":"9d1a984d9a0373b61c0fa8638c4e1bb7953c67da0722712e840ab67b110caa73","schema_version":"1.0","event_id":"sha256:9d1a984d9a0373b61c0fa8638c4e1bb7953c67da0722712e840ab67b110caa73"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX/bundle.json","state_url":"https://pith.science/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX/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":"2026-05-23T00:26:54Z","links":{"resolver":"https://pith.science/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX","bundle":"https://pith.science/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX/bundle.json","state":"https://pith.science/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6Y2QQD5BJVMYCVYASCFP6U5NWX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:6Y2QQD5BJVMYCVYASCFP6U5NWX","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":"a22f32041e7d5c612ac0579ba851ec974f6bb5b3493e0060b613a8338a67dc4f","cross_cats_sorted":["math.NT"],"license":"","primary_cat":"math.CO","submitted_at":"2006-10-29T14:01:46Z","title_canon_sha256":"406900962a4141ff50e28f2a6119a8d673adc63a4b563f97349cc4189b083068"},"schema_version":"1.0","source":{"id":"math/0610893","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610893","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610893v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610893","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"6Y2QQD5BJVMY","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6Y2QQD5BJVMYCVYA","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6Y2QQD5B","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:9d1a984d9a0373b61c0fa8638c4e1bb7953c67da0722712e840ab67b110caa73","target":"graph","created_at":"2026-05-18T01:05:22Z","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":"Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every positive integer m\\leq (k-1)/(n-1) there are more than (k-1)n-(m+1)n(n-1)/2 sets {a_1,...,a_n} such that a_1\\in A_1,..., a_n\\in A_n, and both a_i\\not=a_j and ma_i+b_i\\not=ma_j+b_j (or both ma_i\\not=ma_j and a_i+b_i\\not=a_j+b_j) for all 1\\leq i<j\\leq n.\n This extends a recent result of Dasgupta, K\\'arolyi, Serra and\n Szegedy on Snevily's conjecture. Actually stron","authors_text":"Zhi-Wei Sun","cross_cats":["math.NT"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2006-10-29T14:01:46Z","title":"On Snevily's conjecture and restricted sumsets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610893","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:bbc1fcd7bf84de5ef442c98fddac082a3fd82d6865f8a422998b1b0743901055","target":"record","created_at":"2026-05-18T01:05:22Z","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":"a22f32041e7d5c612ac0579ba851ec974f6bb5b3493e0060b613a8338a67dc4f","cross_cats_sorted":["math.NT"],"license":"","primary_cat":"math.CO","submitted_at":"2006-10-29T14:01:46Z","title_canon_sha256":"406900962a4141ff50e28f2a6119a8d673adc63a4b563f97349cc4189b083068"},"schema_version":"1.0","source":{"id":"math/0610893","kind":"arxiv","version":1}},"canonical_sha256":"f635080fa14d59815700908aff53adb5cc8172f964d03febdae54fe4f1997972","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f635080fa14d59815700908aff53adb5cc8172f964d03febdae54fe4f1997972","first_computed_at":"2026-05-18T01:05:22.533511Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:22.533511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ryOvFtVpNY3XCUlYoDdqFLMQwOAc3riVLrWLA9Bgweizcd5n4PCorhoBPE432P6HBxjiBS3Y7aXNebqBbf2qCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:22.534260Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0610893","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bbc1fcd7bf84de5ef442c98fddac082a3fd82d6865f8a422998b1b0743901055","sha256:9d1a984d9a0373b61c0fa8638c4e1bb7953c67da0722712e840ab67b110caa73"],"state_sha256":"f7b3b1179d941f60b72bea062b387a1448732c1f3d84381224bd5b2fb2bd07eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9ijXMKzh/MxDXB06VkpRE11kFdi//3V1i1Ed/3w9IK7JHhU64MH2OCurL5tDmQbSvg//VXol0UHNzW6BebRBBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T00:26:54.470412Z","bundle_sha256":"377288fa8820deaa5ca90df54687c0b7650cd5bee3be5fc4813c6847695e7315"}}