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The study of $\\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\\rho_{\\pm}(G, m, h)$ equals $\\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\\rho_{\\pm} (G, m, h)$ that we believe gives the exact va"},"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":"1412.1608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-04T10:21:12Z","cross_cats_sorted":[],"title_canon_sha256":"665de0a4c53a1a1e6eb6d90aebdbdaca2e6dfca1b6a95c9e78cb43358cb4bd46","abstract_canon_sha256":"645a8282c27ae1bc2f4b7f6570152c325ab3d93ef597c5d8aa4989b1e3cb9851"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:06.870112Z","signature_b64":"bALiICB+nz5SUcFKTaDubMMMRhRQgQOeRJVe6cp5cmlTlErrTaJDAlmUuqESi/x+VZw+QLF46qCZQRzt7nQ7AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47822ac19603cec8447febda58347b40743d5339335cb3c36e233ba76fec7c0e","last_reissued_at":"2026-05-18T02:32:06.869762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:06.869762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Minimum Size of Signed Sumsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bela Bajnok, Ryan Matzke","submitted_at":"2014-12-04T10:21:12Z","abstract_excerpt":"For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\\rho(G, m, h) = \\min \\{|hA| \\; : \\; A \\subseteq G, |A|=m\\}$$ and $$\\rho_{\\pm} (G, m, h) = \\min \\{|h_{\\pm} A| \\; : \\; A \\subseteq G, |A|=m\\},$$ where $hA$ and $h_{\\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\\rho_{\\pm}(G, m, h)$ equals $\\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\\rho_{\\pm} (G, m, h)$ that we believe gives the exact va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1608","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":"1412.1608","created_at":"2026-05-18T02:32:06.869817+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.1608v1","created_at":"2026-05-18T02:32:06.869817+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1608","created_at":"2026-05-18T02:32:06.869817+00:00"},{"alias_kind":"pith_short_12","alias_value":"I6BCVQMWAPHM","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"I6BCVQMWAPHMQRD7","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"I6BCVQMW","created_at":"2026-05-18T12:28:33.132498+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/I6BCVQMWAPHMQRD75PNFQND3IB","json":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB.json","graph_json":"https://pith.science/api/pith-number/I6BCVQMWAPHMQRD75PNFQND3IB/graph.json","events_json":"https://pith.science/api/pith-number/I6BCVQMWAPHMQRD75PNFQND3IB/events.json","paper":"https://pith.science/paper/I6BCVQMW"},"agent_actions":{"view_html":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB","download_json":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB.json","view_paper":"https://pith.science/paper/I6BCVQMW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.1608&json=true","fetch_graph":"https://pith.science/api/pith-number/I6BCVQMWAPHMQRD75PNFQND3IB/graph.json","fetch_events":"https://pith.science/api/pith-number/I6BCVQMWAPHMQRD75PNFQND3IB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB/action/storage_attestation","attest_author":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB/action/author_attestation","sign_citation":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB/action/citation_signature","submit_replication":"https://pith.science/pith/I6BCVQMWAPHMQRD75PNFQND3IB/action/replication_record"}},"created_at":"2026-05-18T02:32:06.869817+00:00","updated_at":"2026-05-18T02:32:06.869817+00:00"}