{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4HTT6ZOUQ6RTLD6CLI3ZCPN7ES","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":"fe6e244da8cd1e6545ce47f9b7ca6eb99b37227610cbad5e551db33629a5938c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-08-08T09:43:40Z","title_canon_sha256":"bac5be25db26fd8cfe47ce41438fcba013f204f02a2141ab825723f59e13d851"},"schema_version":"1.0","source":{"id":"1708.04309","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.04309","created_at":"2026-05-18T00:37:59Z"},{"alias_kind":"arxiv_version","alias_value":"1708.04309v1","created_at":"2026-05-18T00:37:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.04309","created_at":"2026-05-18T00:37:59Z"},{"alias_kind":"pith_short_12","alias_value":"4HTT6ZOUQ6RT","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"4HTT6ZOUQ6RTLD6C","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"4HTT6ZOU","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:2c69469322cbcc113064ff63b00bb97c21f296f98d16fec69f980f388f7ada05","target":"graph","created_at":"2026-05-18T00:37:59Z","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":"In this paper, we study regular sets in metric measure spaces with bounded Ricci curvature. We prove that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set. Also we define thee dimension of metric measure spaces and prove the lower semicontinuity of that under the Gromov-Hausdorff convergence.","authors_text":"Yu Kitabeppu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-08-08T09:43:40Z","title":"A sufficient condition to a regular set of positive measure on RCD spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.04309","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":"sha256:119d5e41964ee80c518493ed12ce97b2c3d3b843a305bcdb92d880f581d33d11","target":"record","created_at":"2026-05-18T00:37:59Z","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":"fe6e244da8cd1e6545ce47f9b7ca6eb99b37227610cbad5e551db33629a5938c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-08-08T09:43:40Z","title_canon_sha256":"bac5be25db26fd8cfe47ce41438fcba013f204f02a2141ab825723f59e13d851"},"schema_version":"1.0","source":{"id":"1708.04309","kind":"arxiv","version":1}},"canonical_sha256":"e1e73f65d487a3358fc25a37913dbf24948423005419f33adf43c5499f3a2d1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1e73f65d487a3358fc25a37913dbf24948423005419f33adf43c5499f3a2d1a","first_computed_at":"2026-05-18T00:37:59.706855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:37:59.706855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aBwb9xQmrovWaSTr/WIjp2j+I9S8yhxsupv1M6/5hh+AO0uSpQY3KVPjjrK2OR6SfMLaj0cgKizc1tt8SK4vCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:37:59.707237Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.04309","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:119d5e41964ee80c518493ed12ce97b2c3d3b843a305bcdb92d880f581d33d11","sha256:2c69469322cbcc113064ff63b00bb97c21f296f98d16fec69f980f388f7ada05"],"state_sha256":"612b05c40b579d520e39871cf42ba0466b5ce036c7c81979bb85474a46f59460"}