{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X2IVFRG5XFQORP4DMK5WHANFR4","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":"090628b44b3d1020064277ed74771b91b54f7882967cee13ebd1cda42a511468","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-12-15T20:56:40Z","title_canon_sha256":"85ed938d3a55b2a25fcff654d2dd85bf23828e16755d1e63529713682ab476f3"},"schema_version":"1.0","source":{"id":"1512.04936","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04936","created_at":"2026-05-18T01:24:15Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04936v1","created_at":"2026-05-18T01:24:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04936","created_at":"2026-05-18T01:24:15Z"},{"alias_kind":"pith_short_12","alias_value":"X2IVFRG5XFQO","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X2IVFRG5XFQORP4D","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X2IVFRG5","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:ef0ab84892171c6965edf3072093e46fb4481094c7384eca7b86d6fa2622a662","target":"graph","created_at":"2026-05-18T01:24:15Z","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 give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra com","authors_text":"Enrico Le Donne, Severine Rigot","cross_cats":["math.FA","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-12-15T20:56:40Z","title":"Besicovitch Covering Property on graded groups and applications to measure differentiation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04936","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":"sha256:48914756c302111364b64b9c7c3d932587a76e560235f0e353d7bc4371ab245d","target":"record","created_at":"2026-05-18T01:24:15Z","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":"090628b44b3d1020064277ed74771b91b54f7882967cee13ebd1cda42a511468","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-12-15T20:56:40Z","title_canon_sha256":"85ed938d3a55b2a25fcff654d2dd85bf23828e16755d1e63529713682ab476f3"},"schema_version":"1.0","source":{"id":"1512.04936","kind":"arxiv","version":1}},"canonical_sha256":"be9152c4ddb960e8bf8362bb6381a58f291126a47fdff5627b1f4c57884b118b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be9152c4ddb960e8bf8362bb6381a58f291126a47fdff5627b1f4c57884b118b","first_computed_at":"2026-05-18T01:24:15.100730Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:15.100730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g6quKp1Wli/akoyF1V9MkZpSE6tO9Q76VrM0J6SjK06KmvzU/ZiKSxpwNGRr0G0rLtiOgBD3/14AA5rMQHINDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:15.101415Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.04936","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:48914756c302111364b64b9c7c3d932587a76e560235f0e353d7bc4371ab245d","sha256:ef0ab84892171c6965edf3072093e46fb4481094c7384eca7b86d6fa2622a662"],"state_sha256":"bf8d31fad9edc63bd761b2c2014939b293b458dc68bb647bf4a7b859a6d616c4"}