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Suppose that given a sequence $0\\leq f(1)\\leq f(2)\\leq f(3)\\leq \\dots $ such that \\begin{equation}\\sum_{ n=1}^{\\infty} f(n)/n^2 = \\infty.\n  \\end{equation} Then, there exists a sequence $\\{b(n)\\}_{n=1,2,\\dots}$ satisfying \\begin{equation}\\label{eq1} b(n+m) \\leq b(n) + b(m) + f(n+m)\n  \\end{equation} such that the sequence of slopes $\\{ b(n)/n\\}_{n=1,2,\\dots}$ takes every rational number.\n  When the series is bounded we i"},"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":"1810.11723","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-27T22:20:49Z","cross_cats_sorted":[],"title_canon_sha256":"0f757eba943b8fc8135ccc83d2560048ac7c89d2b35be36a7fd8f38f406dccc0","abstract_canon_sha256":"765c3ba4d8dfcfa68f6dd8803a141f18ef769422983924b81faed2c270691351"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:07.631661Z","signature_b64":"6aP994ZWLvmx21swiFNYP1fzbp7+qEVY6vetOHCatNpzOerqubXYpHdn6dLdOCrKO+7Q2EDTOmLvhLjZglddAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f9b13a269ec36a4ca62d42b4d090ca832a78eef76002137078c1ed67e448298","last_reissued_at":"2026-05-18T00:02:07.630988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:07.630988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nearly subadditive sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Imre Z. Ruzsa, Zoltan Furedi","submitted_at":"2018-10-27T22:20:49Z","abstract_excerpt":"We show that the de Bruijn-Erd\\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\\leq f(1)\\leq f(2)\\leq f(3)\\leq \\dots $ such that \\begin{equation}\\sum_{ n=1}^{\\infty} f(n)/n^2 = \\infty.\n  \\end{equation} Then, there exists a sequence $\\{b(n)\\}_{n=1,2,\\dots}$ satisfying \\begin{equation}\\label{eq1} b(n+m) \\leq b(n) + b(m) + f(n+m)\n  \\end{equation} such that the sequence of slopes $\\{ b(n)/n\\}_{n=1,2,\\dots}$ takes every rational number.\n  When the series is bounded we i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.11723","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":"1810.11723","created_at":"2026-05-18T00:02:07.631095+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.11723v1","created_at":"2026-05-18T00:02:07.631095+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.11723","created_at":"2026-05-18T00:02:07.631095+00:00"},{"alias_kind":"pith_short_12","alias_value":"L6NRHITJ5Q3K","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"L6NRHITJ5Q3KJSTC","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"L6NRHITJ","created_at":"2026-05-18T12:32:33.847187+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/L6NRHITJ5Q3KJSTC2QVU2CIMVA","json":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA.json","graph_json":"https://pith.science/api/pith-number/L6NRHITJ5Q3KJSTC2QVU2CIMVA/graph.json","events_json":"https://pith.science/api/pith-number/L6NRHITJ5Q3KJSTC2QVU2CIMVA/events.json","paper":"https://pith.science/paper/L6NRHITJ"},"agent_actions":{"view_html":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA","download_json":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA.json","view_paper":"https://pith.science/paper/L6NRHITJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.11723&json=true","fetch_graph":"https://pith.science/api/pith-number/L6NRHITJ5Q3KJSTC2QVU2CIMVA/graph.json","fetch_events":"https://pith.science/api/pith-number/L6NRHITJ5Q3KJSTC2QVU2CIMVA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/action/storage_attestation","attest_author":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/action/author_attestation","sign_citation":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/action/citation_signature","submit_replication":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/action/replication_record"}},"created_at":"2026-05-18T00:02:07.631095+00:00","updated_at":"2026-05-18T00:02:07.631095+00:00"}