{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7PHGNBTFQJNRDMHN3SAPYLHYWW","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":"c3c6265a01f1a23acac5c9c02c5b8ba31aff932ccc8b2eba9352344fbd8ef1b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-31T05:37:12Z","title_canon_sha256":"e501d80993979a1ddb451b9e3d852b8f05c0f6a7031ac36d533d55121733975e"},"schema_version":"1.0","source":{"id":"1608.08732","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08732","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08732v1","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08732","created_at":"2026-05-18T01:06:40Z"},{"alias_kind":"pith_short_12","alias_value":"7PHGNBTFQJNR","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7PHGNBTFQJNRDMHN","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7PHGNBTF","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:1f01c71fae2463170606f694d17252954557abd01ce3cfd23f2d7158b840f772","target":"graph","created_at":"2026-05-18T01:06:40Z","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 further study the asymptotics of quantization errors for two classes of in-homogeneous self-similar measures $\\mu$. We give a new sufficient condition for the upper quantization coefficient for $\\mu$ to be finite. This, together with our previous work, leads to a necessary and sufficient condition for the upper and lower quantization coefficient of $\\mu$ to be both positive and finite. Furthermore, we determine (estimate) the convergence order of the quantization error in case that the quantization coefficient is infinite.","authors_text":"Sanguo Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-31T05:37:12Z","title":"A note on the quantization error for in-homogeneous self-similar measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08732","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:de125b3dccb8e288d39483286443aeb6ae910102fa75c96c025e504e9a3bc8b1","target":"record","created_at":"2026-05-18T01:06:40Z","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":"c3c6265a01f1a23acac5c9c02c5b8ba31aff932ccc8b2eba9352344fbd8ef1b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-31T05:37:12Z","title_canon_sha256":"e501d80993979a1ddb451b9e3d852b8f05c0f6a7031ac36d533d55121733975e"},"schema_version":"1.0","source":{"id":"1608.08732","kind":"arxiv","version":1}},"canonical_sha256":"fbce668665825b11b0eddc80fc2cf8b5acfa9edaf5c3ac7750a10994794a6f5f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbce668665825b11b0eddc80fc2cf8b5acfa9edaf5c3ac7750a10994794a6f5f","first_computed_at":"2026-05-18T01:06:40.065128Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:40.065128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gTPU1/FknvThu9DqVaIYynERb8WJa1YI4NeZns4ea4Xi4bpU1y7HbtckeQCPgANVKNe+grurSPRD7lBedK7XAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:40.065758Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.08732","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de125b3dccb8e288d39483286443aeb6ae910102fa75c96c025e504e9a3bc8b1","sha256:1f01c71fae2463170606f694d17252954557abd01ce3cfd23f2d7158b840f772"],"state_sha256":"ca023b8f3405ddd879a807f2fdfc6e337031933e539478a1643f105978ec960f"}