{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:2UT7QCDTSQQHZFVWEUKMXYFGFX","short_pith_number":"pith:2UT7QCDT","schema_version":"1.0","canonical_sha256":"d527f8087394207c96b62514cbe0a62dcd49bba68d322e0d5b0b500d6f64861d","source":{"kind":"arxiv","id":"2305.11574","version":1},"attestation_state":"computed","paper":{"title":"On an inverse problem for restricted sumsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xin-Qi Luo, Zhi-Wei Sun","submitted_at":"2023-05-19T10:21:19Z","abstract_excerpt":"Let $n$ be a positive integer, and let $A$ be a set of $k\\ge 2n-1$ integers. For the restricted sumset $$ S_n(A)=\\{a_1+\\cdots +a_n:\\ a_1,\\ldots,a_n\\in A,\\ \\text{and}\\ a_i^2\\neq a_j^2\\ \\text{for} \\ 1\\le i<j\\le n\\}, $$ by a 2002 result of Liu and Sun we have $$|S_n(A)|\\ge (k-1)n-\\frac 32n(n-1)+1.$$ In this paper, we determine the structure of $A$ when the lower bound is attained."},"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":"2305.11574","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-05-19T10:21:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"34ec256ed5dc0ba0daa608a94dde8c74d73c49f2c8db73ee138827aa0914dbff","abstract_canon_sha256":"2ee86d8f3287db39749ca06e0023246ab7d3a6e04c896a7c3295b632ddbbb4b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:11:48.835395Z","signature_b64":"86R3nr54TXzjIzI7D4WG+d3kuPb5SAfu/8HGIUdq+UK9+s5Eyebv/8Q8ekoG8ecLL05NGsAP03ZSkKs9loCcAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d527f8087394207c96b62514cbe0a62dcd49bba68d322e0d5b0b500d6f64861d","last_reissued_at":"2026-07-05T06:11:48.835041Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:11:48.835041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On an inverse problem for restricted sumsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xin-Qi Luo, Zhi-Wei Sun","submitted_at":"2023-05-19T10:21:19Z","abstract_excerpt":"Let $n$ be a positive integer, and let $A$ be a set of $k\\ge 2n-1$ integers. For the restricted sumset $$ S_n(A)=\\{a_1+\\cdots +a_n:\\ a_1,\\ldots,a_n\\in A,\\ \\text{and}\\ a_i^2\\neq a_j^2\\ \\text{for} \\ 1\\le i<j\\le n\\}, $$ by a 2002 result of Liu and Sun we have $$|S_n(A)|\\ge (k-1)n-\\frac 32n(n-1)+1.$$ In this paper, we determine the structure of $A$ when the lower bound is attained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.11574","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.11574/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2305.11574","created_at":"2026-07-05T06:11:48.835097+00:00"},{"alias_kind":"arxiv_version","alias_value":"2305.11574v1","created_at":"2026-07-05T06:11:48.835097+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.11574","created_at":"2026-07-05T06:11:48.835097+00:00"},{"alias_kind":"pith_short_12","alias_value":"2UT7QCDTSQQH","created_at":"2026-07-05T06:11:48.835097+00:00"},{"alias_kind":"pith_short_16","alias_value":"2UT7QCDTSQQHZFVW","created_at":"2026-07-05T06:11:48.835097+00:00"},{"alias_kind":"pith_short_8","alias_value":"2UT7QCDT","created_at":"2026-07-05T06:11:48.835097+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/2UT7QCDTSQQHZFVWEUKMXYFGFX","json":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX.json","graph_json":"https://pith.science/api/pith-number/2UT7QCDTSQQHZFVWEUKMXYFGFX/graph.json","events_json":"https://pith.science/api/pith-number/2UT7QCDTSQQHZFVWEUKMXYFGFX/events.json","paper":"https://pith.science/paper/2UT7QCDT"},"agent_actions":{"view_html":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX","download_json":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX.json","view_paper":"https://pith.science/paper/2UT7QCDT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2305.11574&json=true","fetch_graph":"https://pith.science/api/pith-number/2UT7QCDTSQQHZFVWEUKMXYFGFX/graph.json","fetch_events":"https://pith.science/api/pith-number/2UT7QCDTSQQHZFVWEUKMXYFGFX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX/action/storage_attestation","attest_author":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX/action/author_attestation","sign_citation":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX/action/citation_signature","submit_replication":"https://pith.science/pith/2UT7QCDTSQQHZFVWEUKMXYFGFX/action/replication_record"}},"created_at":"2026-07-05T06:11:48.835097+00:00","updated_at":"2026-07-05T06:11:48.835097+00:00"}