{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:P2REMNDOBZIXUZKPY4BYD5LTB2","short_pith_number":"pith:P2REMNDO","canonical_record":{"source":{"id":"1806.00210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-06-01T06:27:12Z","cross_cats_sorted":[],"title_canon_sha256":"5d8a36e8a791d0d42f4c0ba78b5f93dc69d1d62e41c60ddfa5fc4db967a7012a","abstract_canon_sha256":"8187c35a5444bd44d5bd291f3d8c27743c84add2a7b1ff2e88c23d1ba6818582"},"schema_version":"1.0"},"canonical_sha256":"7ea246346e0e517a654fc70381f5730e8764d41a361b02a99057560fde54fd6c","source":{"kind":"arxiv","id":"1806.00210","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00210","created_at":"2026-05-18T00:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00210v1","created_at":"2026-05-18T00:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00210","created_at":"2026-05-18T00:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"P2REMNDOBZIX","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"P2REMNDOBZIXUZKP","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"P2REMNDO","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:P2REMNDOBZIXUZKPY4BYD5LTB2","target":"record","payload":{"canonical_record":{"source":{"id":"1806.00210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-06-01T06:27:12Z","cross_cats_sorted":[],"title_canon_sha256":"5d8a36e8a791d0d42f4c0ba78b5f93dc69d1d62e41c60ddfa5fc4db967a7012a","abstract_canon_sha256":"8187c35a5444bd44d5bd291f3d8c27743c84add2a7b1ff2e88c23d1ba6818582"},"schema_version":"1.0"},"canonical_sha256":"7ea246346e0e517a654fc70381f5730e8764d41a361b02a99057560fde54fd6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:27.590902Z","signature_b64":"0akPhvKxeR7kaJIponNcsF0aJ2alUSoP9Fx6K+qRdi3l82zTgFaOCPFR02ARAfkDTDg+qEaq6opTCqO5YqeSCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ea246346e0e517a654fc70381f5730e8764d41a361b02a99057560fde54fd6c","last_reissued_at":"2026-05-18T00:14:27.590192Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:27.590192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.00210","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":"2026-05-18T00:14:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5QdYEOvdinXBKPh4SovebIBI5iEhneAta1gkmGMyPz5TIAuW3zKLHJhZN09Z/W2MfAhKI3VJ0JnJJHLGGNUOBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:06:40.531485Z"},"content_sha256":"309a10011504a4cf57f81c21ab2c7c0117a223ecc26faa4a6ea7e29c5268dc09","schema_version":"1.0","event_id":"sha256:309a10011504a4cf57f81c21ab2c7c0117a223ecc26faa4a6ea7e29c5268dc09"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:P2REMNDOBZIXUZKPY4BYD5LTB2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A lemma on the difference quotients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Jianhua Zheng, Kazuya Tohge, Risto Korhonen, Yueyang Zhang","submitted_at":"2018-06-01T06:27:12Z","abstract_excerpt":"Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\\log T(r,f)\\leq r/(\\log r)^{2+\\nu}$, $\\nu>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on difference quotients, where the exceptional set is of finite logarithmic measure. We show the necessity of this set by proving that it must be of infinite linear measure for meromorphic functions whose deficiency is dependent on the choice of the origin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00210","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":"2026-05-18T00:14:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ahoe4h5nG+YsiKF+gi5vkA2v49Y2cOzh+er1dFqAg73gUGDLKqEEnfoBkkUEllTsZps6Dsc3AclvAm5Xnr3+BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:06:40.532161Z"},"content_sha256":"e337f793e1109e26466925f49c1f4941f4451d4cd45e168b104936d81f1609c1","schema_version":"1.0","event_id":"sha256:e337f793e1109e26466925f49c1f4941f4451d4cd45e168b104936d81f1609c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P2REMNDOBZIXUZKPY4BYD5LTB2/bundle.json","state_url":"https://pith.science/pith/P2REMNDOBZIXUZKPY4BYD5LTB2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P2REMNDOBZIXUZKPY4BYD5LTB2/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":"2026-05-26T16:06:40Z","links":{"resolver":"https://pith.science/pith/P2REMNDOBZIXUZKPY4BYD5LTB2","bundle":"https://pith.science/pith/P2REMNDOBZIXUZKPY4BYD5LTB2/bundle.json","state":"https://pith.science/pith/P2REMNDOBZIXUZKPY4BYD5LTB2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P2REMNDOBZIXUZKPY4BYD5LTB2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:P2REMNDOBZIXUZKPY4BYD5LTB2","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":"8187c35a5444bd44d5bd291f3d8c27743c84add2a7b1ff2e88c23d1ba6818582","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-06-01T06:27:12Z","title_canon_sha256":"5d8a36e8a791d0d42f4c0ba78b5f93dc69d1d62e41c60ddfa5fc4db967a7012a"},"schema_version":"1.0","source":{"id":"1806.00210","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00210","created_at":"2026-05-18T00:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00210v1","created_at":"2026-05-18T00:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00210","created_at":"2026-05-18T00:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"P2REMNDOBZIX","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"P2REMNDOBZIXUZKP","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"P2REMNDO","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:e337f793e1109e26466925f49c1f4941f4451d4cd45e168b104936d81f1609c1","target":"graph","created_at":"2026-05-18T00:14:27Z","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":"Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\\log T(r,f)\\leq r/(\\log r)^{2+\\nu}$, $\\nu>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on difference quotients, where the exceptional set is of finite logarithmic measure. We show the necessity of this set by proving that it must be of infinite linear measure for meromorphic functions whose deficiency is dependent on the choice of the origin","authors_text":"Jianhua Zheng, Kazuya Tohge, Risto Korhonen, Yueyang Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-06-01T06:27:12Z","title":"A lemma on the difference quotients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00210","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":"sha256:309a10011504a4cf57f81c21ab2c7c0117a223ecc26faa4a6ea7e29c5268dc09","target":"record","created_at":"2026-05-18T00:14:27Z","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":"8187c35a5444bd44d5bd291f3d8c27743c84add2a7b1ff2e88c23d1ba6818582","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-06-01T06:27:12Z","title_canon_sha256":"5d8a36e8a791d0d42f4c0ba78b5f93dc69d1d62e41c60ddfa5fc4db967a7012a"},"schema_version":"1.0","source":{"id":"1806.00210","kind":"arxiv","version":1}},"canonical_sha256":"7ea246346e0e517a654fc70381f5730e8764d41a361b02a99057560fde54fd6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ea246346e0e517a654fc70381f5730e8764d41a361b02a99057560fde54fd6c","first_computed_at":"2026-05-18T00:14:27.590192Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:27.590192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0akPhvKxeR7kaJIponNcsF0aJ2alUSoP9Fx6K+qRdi3l82zTgFaOCPFR02ARAfkDTDg+qEaq6opTCqO5YqeSCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:27.590902Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.00210","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:309a10011504a4cf57f81c21ab2c7c0117a223ecc26faa4a6ea7e29c5268dc09","sha256:e337f793e1109e26466925f49c1f4941f4451d4cd45e168b104936d81f1609c1"],"state_sha256":"88b1cdfc4ca3849e31d2ac866010beee3eadcaf07d90733bcc9db09407b589e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ftxqAECy1B69xyvI+NFbLq2d1Mv6di83Zf2b7B41jmLq5eb/HqcER3NTgmVrwIf0YLqSbPVrmjQmPqsQ3kUhDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T16:06:40.535628Z","bundle_sha256":"b54cf36feb9197b877ab67dd0f0fd129f38a25d03b64ebdf03b1deb354dcde19"}}