{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:VNEDGLSEMZRVKJ2JML7FRFGQ3L","short_pith_number":"pith:VNEDGLSE","canonical_record":{"source":{"id":"1508.02662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T17:22:18Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"0fa9bea4338f1d2a43103b671dfe6121eaa01ef8430dcd113674abb45f470588","abstract_canon_sha256":"9e46cd02ce3b898e688fefba6dafeb1d6675795df7adbd25fb0d480458c24ab4"},"schema_version":"1.0"},"canonical_sha256":"ab48332e44666355274962fe5894d0daee3357465d02d17495dbdc3bc4faa2f8","source":{"kind":"arxiv","id":"1508.02662","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02662","created_at":"2026-05-18T01:35:28Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02662v1","created_at":"2026-05-18T01:35:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02662","created_at":"2026-05-18T01:35:28Z"},{"alias_kind":"pith_short_12","alias_value":"VNEDGLSEMZRV","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VNEDGLSEMZRVKJ2J","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VNEDGLSE","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:VNEDGLSEMZRVKJ2JML7FRFGQ3L","target":"record","payload":{"canonical_record":{"source":{"id":"1508.02662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T17:22:18Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"0fa9bea4338f1d2a43103b671dfe6121eaa01ef8430dcd113674abb45f470588","abstract_canon_sha256":"9e46cd02ce3b898e688fefba6dafeb1d6675795df7adbd25fb0d480458c24ab4"},"schema_version":"1.0"},"canonical_sha256":"ab48332e44666355274962fe5894d0daee3357465d02d17495dbdc3bc4faa2f8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:28.621965Z","signature_b64":"uNONQYAQSC66vbTg19fzVfHBRa483MW7Q88LG1WQiFngdjLVMT8fu7foJZZZTOVo5iuEmcVhrlT1DIn1i3ufAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab48332e44666355274962fe5894d0daee3357465d02d17495dbdc3bc4faa2f8","last_reissued_at":"2026-05-18T01:35:28.621482Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:28.621482Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.02662","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":"2026-05-18T01:35:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x2WBfYUr20x3C5CVvbicrvdvHkUU+p0uEjxb1pRZ/e+irUrQLcNCOlyWiVVvu1OuQb4V3uhD6sKCujkoyUhwDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:18:32.615265Z"},"content_sha256":"6f26905ac815d416408be671e7433a302746c0781aeb97b4de55bb36871c0fb7","schema_version":"1.0","event_id":"sha256:6f26905ac815d416408be671e7433a302746c0781aeb97b4de55bb36871c0fb7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:VNEDGLSEMZRVKJ2JML7FRFGQ3L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Additive bases in groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.NT","authors_text":"Alain Plagne, Th\\'ai Ho\\`ang L\\^e, Victor Lambert","submitted_at":"2015-08-11T17:22:18Z","abstract_excerpt":"In this paper, we study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions $X$, $S$ and $E$ (defined in the case of the integers) and obtain bounds on them. Our estimates on the functions $S_G$ and $E_G$ are valid for general abelian groups $G$ while in the case of $X_G$ we show that distinct types of behaviours may occur depending on the group $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02662","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":"2026-05-18T01:35:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ry13nJj+jpnwhuVpZFooFUKoW8u6dJeb9wSTSd61hTxy1e0rT/KL8oxT8Brx4I2Vq1DUkr005ayWL3zftop9CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:18:32.615623Z"},"content_sha256":"5fa7c0ba7755e331bbdcc4a8fa75e421445523c3eabac7df345d9254f374d35e","schema_version":"1.0","event_id":"sha256:5fa7c0ba7755e331bbdcc4a8fa75e421445523c3eabac7df345d9254f374d35e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L/bundle.json","state_url":"https://pith.science/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L/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":"2026-06-03T06:18:32Z","links":{"resolver":"https://pith.science/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L","bundle":"https://pith.science/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L/bundle.json","state":"https://pith.science/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VNEDGLSEMZRVKJ2JML7FRFGQ3L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VNEDGLSEMZRVKJ2JML7FRFGQ3L","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":"9e46cd02ce3b898e688fefba6dafeb1d6675795df7adbd25fb0d480458c24ab4","cross_cats_sorted":["math.CO","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T17:22:18Z","title_canon_sha256":"0fa9bea4338f1d2a43103b671dfe6121eaa01ef8430dcd113674abb45f470588"},"schema_version":"1.0","source":{"id":"1508.02662","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02662","created_at":"2026-05-18T01:35:28Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02662v1","created_at":"2026-05-18T01:35:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02662","created_at":"2026-05-18T01:35:28Z"},{"alias_kind":"pith_short_12","alias_value":"VNEDGLSEMZRV","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VNEDGLSEMZRVKJ2J","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VNEDGLSE","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:5fa7c0ba7755e331bbdcc4a8fa75e421445523c3eabac7df345d9254f374d35e","target":"graph","created_at":"2026-05-18T01:35:28Z","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 the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions $X$, $S$ and $E$ (defined in the case of the integers) and obtain bounds on them. Our estimates on the functions $S_G$ and $E_G$ are valid for general abelian groups $G$ while in the case of $X_G$ we show that distinct types of behaviours may occur depending on the group $G$.","authors_text":"Alain Plagne, Th\\'ai Ho\\`ang L\\^e, Victor Lambert","cross_cats":["math.CO","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T17:22:18Z","title":"Additive bases in groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02662","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":"sha256:6f26905ac815d416408be671e7433a302746c0781aeb97b4de55bb36871c0fb7","target":"record","created_at":"2026-05-18T01:35:28Z","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":"9e46cd02ce3b898e688fefba6dafeb1d6675795df7adbd25fb0d480458c24ab4","cross_cats_sorted":["math.CO","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T17:22:18Z","title_canon_sha256":"0fa9bea4338f1d2a43103b671dfe6121eaa01ef8430dcd113674abb45f470588"},"schema_version":"1.0","source":{"id":"1508.02662","kind":"arxiv","version":1}},"canonical_sha256":"ab48332e44666355274962fe5894d0daee3357465d02d17495dbdc3bc4faa2f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab48332e44666355274962fe5894d0daee3357465d02d17495dbdc3bc4faa2f8","first_computed_at":"2026-05-18T01:35:28.621482Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:28.621482Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uNONQYAQSC66vbTg19fzVfHBRa483MW7Q88LG1WQiFngdjLVMT8fu7foJZZZTOVo5iuEmcVhrlT1DIn1i3ufAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:28.621965Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.02662","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f26905ac815d416408be671e7433a302746c0781aeb97b4de55bb36871c0fb7","sha256:5fa7c0ba7755e331bbdcc4a8fa75e421445523c3eabac7df345d9254f374d35e"],"state_sha256":"94252061931c42df7d3cd8c36c02f611ebb7f2287cc2508d72057b9af26d9fb5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fyXQwmrYAeVm61x86JJ02Z2m5MDOtp7MVijE8L6gVJ/cA/nMp27e0ual0CgH6B6+uWyzFRAQKc8DXqlO2Es6Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T06:18:32.617530Z","bundle_sha256":"7a027f32810d5d7d2886e482268c176deb0c57c55d8237d57d5b2539f4845986"}}