{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:24NQNGMI2E5KAMQJ2EIDSYBC2R","short_pith_number":"pith:24NQNGMI","canonical_record":{"source":{"id":"1005.0471","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-04T09:40:27Z","cross_cats_sorted":[],"title_canon_sha256":"d8d1cd09be8534a7cf47efb78ebe9a5bcb1b278d35394f359a308ef1295cf7cc","abstract_canon_sha256":"fa3a37c69ba5cf62afaebc1ebab0205fe0ebb9ae30c814428e39eb7cde079e61"},"schema_version":"1.0"},"canonical_sha256":"d71b069988d13aa03209d110396022d47ab859d85bfb4c25e8da311d0c01754c","source":{"kind":"arxiv","id":"1005.0471","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0471","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0471v3","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0471","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"24NQNGMI2E5K","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"24NQNGMI2E5KAMQJ","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"24NQNGMI","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:24NQNGMI2E5KAMQJ2EIDSYBC2R","target":"record","payload":{"canonical_record":{"source":{"id":"1005.0471","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-04T09:40:27Z","cross_cats_sorted":[],"title_canon_sha256":"d8d1cd09be8534a7cf47efb78ebe9a5bcb1b278d35394f359a308ef1295cf7cc","abstract_canon_sha256":"fa3a37c69ba5cf62afaebc1ebab0205fe0ebb9ae30c814428e39eb7cde079e61"},"schema_version":"1.0"},"canonical_sha256":"d71b069988d13aa03209d110396022d47ab859d85bfb4c25e8da311d0c01754c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.470076Z","signature_b64":"X3jSfvrOnr55uRXV+X4KKDnw8rPFeMvKjIzkD9KYGloCR/O1jX3HYz7SKQZVkCijr49ku1iHouLM+4DH8OweBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d71b069988d13aa03209d110396022d47ab859d85bfb4c25e8da311d0c01754c","last_reissued_at":"2026-05-18T03:07:02.469451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.469451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.0471","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5YUsQFDl3yTeJvSxAHbr7VfkBu+VvQAzuHvp5+p+LKU55W3WlH1sD7RLCypLCbizDSUpbhn35lnjpDdqLRAhAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:51:30.016790Z"},"content_sha256":"08a2c5cfc52027fd82a3ed4336e1f8af9c5032f0b2e5cb1f8be1357c6ea183c3","schema_version":"1.0","event_id":"sha256:08a2c5cfc52027fd82a3ed4336e1f8af9c5032f0b2e5cb1f8be1357c6ea183c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:24NQNGMI2E5KAMQJ2EIDSYBC2R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A quantitative version of Steinhaus' theorem for compact, connected, rank-one symmetric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fernando M\\'ario de Oliveira Filho, Frank Vallentin","submitted_at":"2010-05-04T09:40:27Z","abstract_excerpt":"Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$, $d_2$, ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0471","kind":"arxiv","version":3},"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":"2026-05-18T03:07:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VnaGHFrZrbVTt3hOJq065s9rvHaMuhRzuW0E0fqOkqgZQ2N18tjh/lPMRdMIXBRBfhPvCJEoc6Zg6N8xcgOcBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:51:30.017605Z"},"content_sha256":"3259cc3e276d062a2a3c74bd65819bfb1010e0f00ba969a6b81c70e7b0b9edbe","schema_version":"1.0","event_id":"sha256:3259cc3e276d062a2a3c74bd65819bfb1010e0f00ba969a6b81c70e7b0b9edbe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R/bundle.json","state_url":"https://pith.science/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R/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":"2026-06-10T23:51:30Z","links":{"resolver":"https://pith.science/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R","bundle":"https://pith.science/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R/bundle.json","state":"https://pith.science/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/24NQNGMI2E5KAMQJ2EIDSYBC2R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:24NQNGMI2E5KAMQJ2EIDSYBC2R","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":"fa3a37c69ba5cf62afaebc1ebab0205fe0ebb9ae30c814428e39eb7cde079e61","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-04T09:40:27Z","title_canon_sha256":"d8d1cd09be8534a7cf47efb78ebe9a5bcb1b278d35394f359a308ef1295cf7cc"},"schema_version":"1.0","source":{"id":"1005.0471","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0471","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0471v3","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0471","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"24NQNGMI2E5K","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"24NQNGMI2E5KAMQJ","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"24NQNGMI","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:3259cc3e276d062a2a3c74bd65819bfb1010e0f00ba969a6b81c70e7b0b9edbe","target":"graph","created_at":"2026-05-18T03:07:02Z","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":"Let $d_1$, $d_2$, ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus' theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances $d_1$, $d_2$, ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances.","authors_text":"Fernando M\\'ario de Oliveira Filho, Frank Vallentin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-04T09:40:27Z","title":"A quantitative version of Steinhaus' theorem for compact, connected, rank-one symmetric spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0471","kind":"arxiv","version":3},"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":"sha256:08a2c5cfc52027fd82a3ed4336e1f8af9c5032f0b2e5cb1f8be1357c6ea183c3","target":"record","created_at":"2026-05-18T03:07:02Z","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":"fa3a37c69ba5cf62afaebc1ebab0205fe0ebb9ae30c814428e39eb7cde079e61","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-05-04T09:40:27Z","title_canon_sha256":"d8d1cd09be8534a7cf47efb78ebe9a5bcb1b278d35394f359a308ef1295cf7cc"},"schema_version":"1.0","source":{"id":"1005.0471","kind":"arxiv","version":3}},"canonical_sha256":"d71b069988d13aa03209d110396022d47ab859d85bfb4c25e8da311d0c01754c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d71b069988d13aa03209d110396022d47ab859d85bfb4c25e8da311d0c01754c","first_computed_at":"2026-05-18T03:07:02.469451Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:02.469451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X3jSfvrOnr55uRXV+X4KKDnw8rPFeMvKjIzkD9KYGloCR/O1jX3HYz7SKQZVkCijr49ku1iHouLM+4DH8OweBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:02.470076Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.0471","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:08a2c5cfc52027fd82a3ed4336e1f8af9c5032f0b2e5cb1f8be1357c6ea183c3","sha256:3259cc3e276d062a2a3c74bd65819bfb1010e0f00ba969a6b81c70e7b0b9edbe"],"state_sha256":"2155cbd61cd8ef2ccce1ab85a2bf1ed51ed9ca4194f35f4cf8c84bc8c85a1ff0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"13SsmxEIF/K8Yx8Q4QAGXkv7KNvSqijNAZoqEu3wd1QkP9zInIEvdhgSDwQ0htpGRq2ytIJ/PydUhNlCz6D2CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:51:30.021954Z","bundle_sha256":"531c70f14e1d0f21f39c9100c5f2b126f3f567851d3df9701f7f94ba5321a8e1"}}