{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:L6NRHITJ5Q3KJSTC2QVU2CIMVA","short_pith_number":"pith:L6NRHITJ","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"},"canonical_sha256":"5f9b13a269ec36a4ca62d42b4d090ca832a78eef76002137078c1ed67e448298","source":{"kind":"arxiv","id":"1810.11723","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.11723","created_at":"2026-05-18T00:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"1810.11723v1","created_at":"2026-05-18T00:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.11723","created_at":"2026-05-18T00:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"L6NRHITJ5Q3K","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"L6NRHITJ5Q3KJSTC","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"L6NRHITJ","created_at":"2026-05-18T12:32:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:L6NRHITJ5Q3KJSTC2QVU2CIMVA","target":"record","payload":{"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"},"canonical_sha256":"5f9b13a269ec36a4ca62d42b4d090ca832a78eef76002137078c1ed67e448298","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"},"source_kind":"arxiv","source_id":"1810.11723","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:02:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jPPNTFWqtMPF8mP1lHJ8ibwb2ysRwde8nhe7i3sA55/kj+DL3cAjC2g3YVzLxSRPOaRhqO+u4PghEY1nK986Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:32:27.938820Z"},"content_sha256":"686d3a14c1c26e4e473afea33a4afe272c3c542f11c79e3d3d7e00b764463e5c","schema_version":"1.0","event_id":"sha256:686d3a14c1c26e4e473afea33a4afe272c3c542f11c79e3d3d7e00b764463e5c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:L6NRHITJ5Q3KJSTC2QVU2CIMVA","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:02:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FJLJtwpoL6nkMQAkjXZgP31pi5WHu6Kii8eORsglxgipa4d8Y9bMyf7P9rt1pBQgTftaezdToJNXIYCj1EIdDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:32:27.939509Z"},"content_sha256":"0c85952c05f31071def6e9e7bff87863113add53ea79625ff438c0ddbb31185e","schema_version":"1.0","event_id":"sha256:0c85952c05f31071def6e9e7bff87863113add53ea79625ff438c0ddbb31185e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/bundle.json","state_url":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T09:32:27Z","links":{"resolver":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA","bundle":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/bundle.json","state":"https://pith.science/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L6NRHITJ5Q3KJSTC2QVU2CIMVA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:L6NRHITJ5Q3KJSTC2QVU2CIMVA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"765c3ba4d8dfcfa68f6dd8803a141f18ef769422983924b81faed2c270691351","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-27T22:20:49Z","title_canon_sha256":"0f757eba943b8fc8135ccc83d2560048ac7c89d2b35be36a7fd8f38f406dccc0"},"schema_version":"1.0","source":{"id":"1810.11723","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.11723","created_at":"2026-05-18T00:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"1810.11723v1","created_at":"2026-05-18T00:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.11723","created_at":"2026-05-18T00:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"L6NRHITJ5Q3K","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"L6NRHITJ5Q3KJSTC","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"L6NRHITJ","created_at":"2026-05-18T12:32:33Z"}],"graph_snapshots":[{"event_id":"sha256:0c85952c05f31071def6e9e7bff87863113add53ea79625ff438c0ddbb31185e","target":"graph","created_at":"2026-05-18T00:02:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Imre Z. Ruzsa, Zoltan Furedi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-27T22:20:49Z","title":"Nearly subadditive sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.11723","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:686d3a14c1c26e4e473afea33a4afe272c3c542f11c79e3d3d7e00b764463e5c","target":"record","created_at":"2026-05-18T00:02:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"765c3ba4d8dfcfa68f6dd8803a141f18ef769422983924b81faed2c270691351","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-27T22:20:49Z","title_canon_sha256":"0f757eba943b8fc8135ccc83d2560048ac7c89d2b35be36a7fd8f38f406dccc0"},"schema_version":"1.0","source":{"id":"1810.11723","kind":"arxiv","version":1}},"canonical_sha256":"5f9b13a269ec36a4ca62d42b4d090ca832a78eef76002137078c1ed67e448298","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f9b13a269ec36a4ca62d42b4d090ca832a78eef76002137078c1ed67e448298","first_computed_at":"2026-05-18T00:02:07.630988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:07.630988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6aP994ZWLvmx21swiFNYP1fzbp7+qEVY6vetOHCatNpzOerqubXYpHdn6dLdOCrKO+7Q2EDTOmLvhLjZglddAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:07.631661Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.11723","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:686d3a14c1c26e4e473afea33a4afe272c3c542f11c79e3d3d7e00b764463e5c","sha256:0c85952c05f31071def6e9e7bff87863113add53ea79625ff438c0ddbb31185e"],"state_sha256":"a128448c2ec8430819416a35001087c3d85996e35acf9856b3cf2d6eb62f19c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z6lsMVH2QYSRh9zQEBVz3KdojLI0xokUah/UgZNeHp+7Ul+4pBVy7c45YAEV6U1ts1zrB8bEvcuU/1WBYjoACA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T09:32:27.943425Z","bundle_sha256":"520dcce0b2dc906d1694497b1b42270365352bd38826d62b79b8826456cf33ee"}}