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Nathanson proved that $\\tilde{C}_r(A)$ is always at most $r+1$; here we extend this result to prove that $\\tilde{C}_r(A)$ is always at least $r$, and determine all sets $A$ for which $\\tilde{C}_r(A)=r$."},"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":"1603.06553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-21T19:37:20Z","cross_cats_sorted":[],"title_canon_sha256":"60159bef8d0860155b41ec6dbcca379ee91bb7bdc3d7547dab785f6982624385","abstract_canon_sha256":"bf1ad54eecb85769ac8ce969eb9d85c8c4c4b56fc91834210050b4e383fd328d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:48.294168Z","signature_b64":"h6gwa7r0CnUXvCeTKpPniMDs+fCm8an0SE3sIyz+i6xWFhAAw8esLeX0ltTrSDM9KWw77+Y5yQqSZ1gWCTMvDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"13a5ffcbd5880013e037083f9f564d90686bd39bc34d6e9a7addffd7d54081eb","last_reissued_at":"2026-05-18T01:18:48.293677Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:48.293677Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Asymptotic Approximate Groups of Integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bela Bajnok","submitted_at":"2016-03-21T19:37:20Z","abstract_excerpt":"Let $r$ be a positive integer, and let $A$ be a nonempty finite set of at least two integers. 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Nathanson proved that $\\tilde{C}_r(A)$ is always at most $r+1$; here we extend this result to prove that $\\tilde{C}_r(A)$ is always at least $r$, and determine all sets $A$ for which $\\tilde{C}_r(A)=r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06553","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":"1603.06553","created_at":"2026-05-18T01:18:48.293757+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.06553v1","created_at":"2026-05-18T01:18:48.293757+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06553","created_at":"2026-05-18T01:18:48.293757+00:00"},{"alias_kind":"pith_short_12","alias_value":"COS77S6VRAAB","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"COS77S6VRAABHYBX","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"COS77S6V","created_at":"2026-05-18T12:30:09.641336+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/COS77S6VRAABHYBXBA7Z6VSNSB","json":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB.json","graph_json":"https://pith.science/api/pith-number/COS77S6VRAABHYBXBA7Z6VSNSB/graph.json","events_json":"https://pith.science/api/pith-number/COS77S6VRAABHYBXBA7Z6VSNSB/events.json","paper":"https://pith.science/paper/COS77S6V"},"agent_actions":{"view_html":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB","download_json":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB.json","view_paper":"https://pith.science/paper/COS77S6V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.06553&json=true","fetch_graph":"https://pith.science/api/pith-number/COS77S6VRAABHYBXBA7Z6VSNSB/graph.json","fetch_events":"https://pith.science/api/pith-number/COS77S6VRAABHYBXBA7Z6VSNSB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB/action/storage_attestation","attest_author":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB/action/author_attestation","sign_citation":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB/action/citation_signature","submit_replication":"https://pith.science/pith/COS77S6VRAABHYBXBA7Z6VSNSB/action/replication_record"}},"created_at":"2026-05-18T01:18:48.293757+00:00","updated_at":"2026-05-18T01:18:48.293757+00:00"}