{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:GI3MZYDUD6VOOZ7JH7M3BH7VTL","short_pith_number":"pith:GI3MZYDU","schema_version":"1.0","canonical_sha256":"3236cce0741faae767e93fd9b09ff59ac1c9bfe85e0589e6e312a7b8fd34f5ba","source":{"kind":"arxiv","id":"1011.1797","version":1},"attestation_state":"computed","paper":{"title":"Topology of Cayley Graphs Applied to Inverse Additive Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Yahya Ould Hamidoune","submitted_at":"2010-11-08T13:25:24Z","abstract_excerpt":"We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\\le r \\le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....)."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1011.1797","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-08T13:25:24Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"32578a75ceac0cdf1ab4fddeab74ee6279493136660f76d4d3a308641a51b00f","abstract_canon_sha256":"d7ed527a006ba19a145405944570a8a93f45e7973343146e426dced48edc793a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:49.870829Z","signature_b64":"rn75INpEzyMIlkSqkYYHSk/IG1KtlP9F+qJfZ4ebRX6WRRmZo8aYXay1RCxho0mRNlOgGiZcuKSNgdu0Ba+ZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3236cce0741faae767e93fd9b09ff59ac1c9bfe85e0589e6e312a7b8fd34f5ba","last_reissued_at":"2026-05-18T04:36:49.870292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:49.870292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topology of Cayley Graphs Applied to Inverse Additive Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Yahya Ould Hamidoune","submitted_at":"2010-11-08T13:25:24Z","abstract_excerpt":"We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\\le r \\le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1797","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1011.1797","created_at":"2026-05-18T04:36:49.870376+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.1797v1","created_at":"2026-05-18T04:36:49.870376+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1797","created_at":"2026-05-18T04:36:49.870376+00:00"},{"alias_kind":"pith_short_12","alias_value":"GI3MZYDUD6VO","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"GI3MZYDUD6VOOZ7J","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"GI3MZYDU","created_at":"2026-05-18T12:26:07.630475+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL","json":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL.json","graph_json":"https://pith.science/api/pith-number/GI3MZYDUD6VOOZ7JH7M3BH7VTL/graph.json","events_json":"https://pith.science/api/pith-number/GI3MZYDUD6VOOZ7JH7M3BH7VTL/events.json","paper":"https://pith.science/paper/GI3MZYDU"},"agent_actions":{"view_html":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL","download_json":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL.json","view_paper":"https://pith.science/paper/GI3MZYDU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.1797&json=true","fetch_graph":"https://pith.science/api/pith-number/GI3MZYDUD6VOOZ7JH7M3BH7VTL/graph.json","fetch_events":"https://pith.science/api/pith-number/GI3MZYDUD6VOOZ7JH7M3BH7VTL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL/action/storage_attestation","attest_author":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL/action/author_attestation","sign_citation":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL/action/citation_signature","submit_replication":"https://pith.science/pith/GI3MZYDUD6VOOZ7JH7M3BH7VTL/action/replication_record"}},"created_at":"2026-05-18T04:36:49.870376+00:00","updated_at":"2026-05-18T04:36:49.870376+00:00"}