{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XM5RWVASBB33GFD6YY55IGRH74","short_pith_number":"pith:XM5RWVAS","canonical_record":{"source":{"id":"1312.2971","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-12-10T21:30:05Z","cross_cats_sorted":[],"title_canon_sha256":"73dfe1d754e20557d822d05b333fba264d63675e0659702c0a50f0d1adcee54a","abstract_canon_sha256":"185eee4d6c5d9ce764f8e4ab5a982278b2a25cc380d5d7315bd140f7f5c8b67c"},"schema_version":"1.0"},"canonical_sha256":"bb3b1b54120877b3147ec63bd41a27ff345705c5d16abdcc2d785f56b1128c23","source":{"kind":"arxiv","id":"1312.2971","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.2971","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"1312.2971v1","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2971","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"XM5RWVASBB33","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XM5RWVASBB33GFD6","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XM5RWVAS","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XM5RWVASBB33GFD6YY55IGRH74","target":"record","payload":{"canonical_record":{"source":{"id":"1312.2971","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-12-10T21:30:05Z","cross_cats_sorted":[],"title_canon_sha256":"73dfe1d754e20557d822d05b333fba264d63675e0659702c0a50f0d1adcee54a","abstract_canon_sha256":"185eee4d6c5d9ce764f8e4ab5a982278b2a25cc380d5d7315bd140f7f5c8b67c"},"schema_version":"1.0"},"canonical_sha256":"bb3b1b54120877b3147ec63bd41a27ff345705c5d16abdcc2d785f56b1128c23","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:00.879136Z","signature_b64":"7ApJVdZHTIHlkp+2DjtHv/3QXqemulzXGmSa1aD22Mz0uZHwlqMjb7dhUqjrH+fi0rR8QhNn/tfFb0GW68ltCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb3b1b54120877b3147ec63bd41a27ff345705c5d16abdcc2d785f56b1128c23","last_reissued_at":"2026-05-18T03:05:00.878545Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:00.878545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.2971","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":"2026-05-18T03:05:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FhzT2Ej1ECYLdtDUFM9JmJrHNwkjpUwCbJh8+HYlFnL8+sNjQUM1S8Ij2drRvm3sUhHK9LicZiguW0E692hLDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:24:33.735515Z"},"content_sha256":"939ce1a30fc94906054456bd9961cb25e56bde83701bc3ab184ddb17de078421","schema_version":"1.0","event_id":"sha256:939ce1a30fc94906054456bd9961cb25e56bde83701bc3ab184ddb17de078421"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XM5RWVASBB33GFD6YY55IGRH74","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On s-sets in spaces of homogeneous type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Marilina Carena, Marisa Toschi","submitted_at":"2013-12-10T21:30:05Z","abstract_excerpt":"Let $(X,d,\\mu)$ be a space of homogeneous type. In this note we study the relationship between two types of $s$-sets: relative to a distance and relative to a measure. We find a condition on a closed subset $F$ of $X$ under which we have that $F$ is $s$-set relative to the measure $\\mu$ if and only if $F$ is $s$-set relative to $\\delta$. Here $\\delta$ denotes the quasi-distance defined by Mac\\'ias and Segovia such that $(X,\\delta,\\mu)$ is a normal space. In order to prove this result, we show a covering type lemma and a type of Hausdorff measure based criteria for the $s$-set condition relativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2971","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":"2026-05-18T03:05:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VgrwoVKpeMgfDJ0VfukQ3GF1WGvKO/jISfr3uCIUZuoGoLGRtjgafwtSZ0ANMdZmY5kWSdYKRizWOAEks2NTBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:24:33.736214Z"},"content_sha256":"b2a475c9245e5e090787c74e4f3dcd43f99fe81c672d946fac88569b2a579184","schema_version":"1.0","event_id":"sha256:b2a475c9245e5e090787c74e4f3dcd43f99fe81c672d946fac88569b2a579184"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XM5RWVASBB33GFD6YY55IGRH74/bundle.json","state_url":"https://pith.science/pith/XM5RWVASBB33GFD6YY55IGRH74/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XM5RWVASBB33GFD6YY55IGRH74/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":"2026-06-10T22:24:33Z","links":{"resolver":"https://pith.science/pith/XM5RWVASBB33GFD6YY55IGRH74","bundle":"https://pith.science/pith/XM5RWVASBB33GFD6YY55IGRH74/bundle.json","state":"https://pith.science/pith/XM5RWVASBB33GFD6YY55IGRH74/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XM5RWVASBB33GFD6YY55IGRH74/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XM5RWVASBB33GFD6YY55IGRH74","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":"185eee4d6c5d9ce764f8e4ab5a982278b2a25cc380d5d7315bd140f7f5c8b67c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-12-10T21:30:05Z","title_canon_sha256":"73dfe1d754e20557d822d05b333fba264d63675e0659702c0a50f0d1adcee54a"},"schema_version":"1.0","source":{"id":"1312.2971","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.2971","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"1312.2971v1","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2971","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"XM5RWVASBB33","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XM5RWVASBB33GFD6","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XM5RWVAS","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:b2a475c9245e5e090787c74e4f3dcd43f99fe81c672d946fac88569b2a579184","target":"graph","created_at":"2026-05-18T03:05:00Z","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 $(X,d,\\mu)$ be a space of homogeneous type. In this note we study the relationship between two types of $s$-sets: relative to a distance and relative to a measure. We find a condition on a closed subset $F$ of $X$ under which we have that $F$ is $s$-set relative to the measure $\\mu$ if and only if $F$ is $s$-set relative to $\\delta$. Here $\\delta$ denotes the quasi-distance defined by Mac\\'ias and Segovia such that $(X,\\delta,\\mu)$ is a normal space. In order to prove this result, we show a covering type lemma and a type of Hausdorff measure based criteria for the $s$-set condition relativ","authors_text":"Marilina Carena, Marisa Toschi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-12-10T21:30:05Z","title":"On s-sets in spaces of homogeneous type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2971","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":"sha256:939ce1a30fc94906054456bd9961cb25e56bde83701bc3ab184ddb17de078421","target":"record","created_at":"2026-05-18T03:05:00Z","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":"185eee4d6c5d9ce764f8e4ab5a982278b2a25cc380d5d7315bd140f7f5c8b67c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-12-10T21:30:05Z","title_canon_sha256":"73dfe1d754e20557d822d05b333fba264d63675e0659702c0a50f0d1adcee54a"},"schema_version":"1.0","source":{"id":"1312.2971","kind":"arxiv","version":1}},"canonical_sha256":"bb3b1b54120877b3147ec63bd41a27ff345705c5d16abdcc2d785f56b1128c23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb3b1b54120877b3147ec63bd41a27ff345705c5d16abdcc2d785f56b1128c23","first_computed_at":"2026-05-18T03:05:00.878545Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:00.878545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7ApJVdZHTIHlkp+2DjtHv/3QXqemulzXGmSa1aD22Mz0uZHwlqMjb7dhUqjrH+fi0rR8QhNn/tfFb0GW68ltCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:00.879136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.2971","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:939ce1a30fc94906054456bd9961cb25e56bde83701bc3ab184ddb17de078421","sha256:b2a475c9245e5e090787c74e4f3dcd43f99fe81c672d946fac88569b2a579184"],"state_sha256":"9cf187c8f2d02993e210a30287bff01f1c8f855358f0ff0ddace87a801b80a53"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yan7sPr9XIz5fcEHj8GlHzwEbjg8Rf8dmN/Lr5TX8wD9QBSAB40dOqaYNcu73WcajsvdYdD0F2prUqxfr7XKDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T22:24:33.740688Z","bundle_sha256":"97bd09865777212c02c70377c22d1e7f7ff1a51af725acb89c3d44dd99df623a"}}