{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:OLVZO7L7QUWMIKS6GJNF7FHYIV","short_pith_number":"pith:OLVZO7L7","canonical_record":{"source":{"id":"1210.0816","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T16:00:31Z","cross_cats_sorted":["math.GT","math.NT"],"title_canon_sha256":"d317ec34912cc53ac974be60db0e1185a36c3f9725dbea12a7b142b93af71bd2","abstract_canon_sha256":"83c4631ea9442593f5a28cbbf47d0837c61b880da5539b30f8e887c150121cff"},"schema_version":"1.0"},"canonical_sha256":"72eb977d7f852cc42a5e325a5f94f845505f210fb271c3b98dd165a92829170b","source":{"kind":"arxiv","id":"1210.0816","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0816","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0816v2","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0816","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"pith_short_12","alias_value":"OLVZO7L7QUWM","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"OLVZO7L7QUWMIKS6","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"OLVZO7L7","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:OLVZO7L7QUWMIKS6GJNF7FHYIV","target":"record","payload":{"canonical_record":{"source":{"id":"1210.0816","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T16:00:31Z","cross_cats_sorted":["math.GT","math.NT"],"title_canon_sha256":"d317ec34912cc53ac974be60db0e1185a36c3f9725dbea12a7b142b93af71bd2","abstract_canon_sha256":"83c4631ea9442593f5a28cbbf47d0837c61b880da5539b30f8e887c150121cff"},"schema_version":"1.0"},"canonical_sha256":"72eb977d7f852cc42a5e325a5f94f845505f210fb271c3b98dd165a92829170b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:13.304681Z","signature_b64":"ao7msesZa0UtN9b9+wrQlDr17x7gOzPeo/Yxp3vFb+M+asKnLRowDPNTx4mxQWb8S8gmCKcS3BS7JdujVHIdBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72eb977d7f852cc42a5e325a5f94f845505f210fb271c3b98dd165a92829170b","last_reissued_at":"2026-05-18T02:42:13.304199Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:13.304199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.0816","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:42:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IGDW5RorJiS4KSGgR+dTGJtiAJnffbVb0xTqqBQNh1UjyzsgUFqgUE7x3ZvkMEmB7fPQRI1Ak4hEcEI/CfDNBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:24:06.455293Z"},"content_sha256":"583715e922459e745c1da09f6b88fc3c87692610f760a63ac196c7bf84e480b8","schema_version":"1.0","event_id":"sha256:583715e922459e745c1da09f6b88fc3c87692610f760a63ac196c7bf84e480b8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:OLVZO7L7QUWMIKS6GJNF7FHYIV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gap distributions and homogeneous dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.NT"],"primary_cat":"math.DS","authors_text":"Jayadev S. Athreya","submitted_at":"2012-10-02T16:00:31Z","abstract_excerpt":"We survey the use of dynamics of $SL(2, \\R)$-actions to understand gap distributions for various sequences of subsets of $[0, 1)$, particularly those arising from special trajectories of various two-dimensional dynamical systems. We state and prove an abstract theorem that gives a unified explanation for some of the examples we present."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0816","kind":"arxiv","version":2},"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":"2026-05-18T02:42:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4KTysLIysbwv6EtUANH8xX47qrW8zXqtWC3XJLLx1z2Nf7M1rxYUKjbCVU211JGw6Aw4SWlTPKfjpImC3oW2DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:24:06.455918Z"},"content_sha256":"6f84717d60fdab1d5464a5ef8661859ef1ba332790f437cd8c1e2576849d6440","schema_version":"1.0","event_id":"sha256:6f84717d60fdab1d5464a5ef8661859ef1ba332790f437cd8c1e2576849d6440"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV/bundle.json","state_url":"https://pith.science/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV/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":"2026-06-02T22:24:06Z","links":{"resolver":"https://pith.science/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV","bundle":"https://pith.science/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV/bundle.json","state":"https://pith.science/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OLVZO7L7QUWMIKS6GJNF7FHYIV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:OLVZO7L7QUWMIKS6GJNF7FHYIV","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":"83c4631ea9442593f5a28cbbf47d0837c61b880da5539b30f8e887c150121cff","cross_cats_sorted":["math.GT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T16:00:31Z","title_canon_sha256":"d317ec34912cc53ac974be60db0e1185a36c3f9725dbea12a7b142b93af71bd2"},"schema_version":"1.0","source":{"id":"1210.0816","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0816","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0816v2","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0816","created_at":"2026-05-18T02:42:13Z"},{"alias_kind":"pith_short_12","alias_value":"OLVZO7L7QUWM","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"OLVZO7L7QUWMIKS6","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"OLVZO7L7","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:6f84717d60fdab1d5464a5ef8661859ef1ba332790f437cd8c1e2576849d6440","target":"graph","created_at":"2026-05-18T02:42:13Z","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 survey the use of dynamics of $SL(2, \\R)$-actions to understand gap distributions for various sequences of subsets of $[0, 1)$, particularly those arising from special trajectories of various two-dimensional dynamical systems. We state and prove an abstract theorem that gives a unified explanation for some of the examples we present.","authors_text":"Jayadev S. Athreya","cross_cats":["math.GT","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T16:00:31Z","title":"Gap distributions and homogeneous dynamics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0816","kind":"arxiv","version":2},"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":"sha256:583715e922459e745c1da09f6b88fc3c87692610f760a63ac196c7bf84e480b8","target":"record","created_at":"2026-05-18T02:42:13Z","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":"83c4631ea9442593f5a28cbbf47d0837c61b880da5539b30f8e887c150121cff","cross_cats_sorted":["math.GT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T16:00:31Z","title_canon_sha256":"d317ec34912cc53ac974be60db0e1185a36c3f9725dbea12a7b142b93af71bd2"},"schema_version":"1.0","source":{"id":"1210.0816","kind":"arxiv","version":2}},"canonical_sha256":"72eb977d7f852cc42a5e325a5f94f845505f210fb271c3b98dd165a92829170b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"72eb977d7f852cc42a5e325a5f94f845505f210fb271c3b98dd165a92829170b","first_computed_at":"2026-05-18T02:42:13.304199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:13.304199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ao7msesZa0UtN9b9+wrQlDr17x7gOzPeo/Yxp3vFb+M+asKnLRowDPNTx4mxQWb8S8gmCKcS3BS7JdujVHIdBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:13.304681Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.0816","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:583715e922459e745c1da09f6b88fc3c87692610f760a63ac196c7bf84e480b8","sha256:6f84717d60fdab1d5464a5ef8661859ef1ba332790f437cd8c1e2576849d6440"],"state_sha256":"167fc2d2c3fcdb90402d3c4dcf240b72b833ec1b3849fe0481bb8e0e3cb9644e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8I4mCDQZI6fS6tUIiJL9RO/xb47fj95rfcLgWysCB2Yvsl3hysbRWtlQQZaTzMcLFFuTNlVPdXjIbyMVJ/NSDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:24:06.458016Z","bundle_sha256":"05a0e3ccb5ff4079cd099a51af94eb02cd57b231aea61e23dfdfcf5b1cf29636"}}