{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:Z3ZDU2QWZEYNMPK2XWEL63KWXW","short_pith_number":"pith:Z3ZDU2QW","canonical_record":{"source":{"id":"1009.5076","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-26T09:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"591a9a1e80aa4f3499e276eca154696b403c82f53634c12d48737a11fd72bc9f","abstract_canon_sha256":"b8464f40d49e64722f29b7306ef4664d02be8d2b6565fe60ed27dd566a9ab85c"},"schema_version":"1.0"},"canonical_sha256":"cef23a6a16c930d63d5abd88bf6d56bdbfc457628d8a17398056f73b3f43cb3b","source":{"kind":"arxiv","id":"1009.5076","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.5076","created_at":"2026-05-18T04:40:19Z"},{"alias_kind":"arxiv_version","alias_value":"1009.5076v1","created_at":"2026-05-18T04:40:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5076","created_at":"2026-05-18T04:40:19Z"},{"alias_kind":"pith_short_12","alias_value":"Z3ZDU2QWZEYN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Z3ZDU2QWZEYNMPK2","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Z3ZDU2QW","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:Z3ZDU2QWZEYNMPK2XWEL63KWXW","target":"record","payload":{"canonical_record":{"source":{"id":"1009.5076","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-26T09:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"591a9a1e80aa4f3499e276eca154696b403c82f53634c12d48737a11fd72bc9f","abstract_canon_sha256":"b8464f40d49e64722f29b7306ef4664d02be8d2b6565fe60ed27dd566a9ab85c"},"schema_version":"1.0"},"canonical_sha256":"cef23a6a16c930d63d5abd88bf6d56bdbfc457628d8a17398056f73b3f43cb3b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:19.262488Z","signature_b64":"N5G3JnklhQ1Mhf4/n5XgaIRSP0atnJOWFHP9B68CJTIa/HAikoCP79j4Ni7WC/z03txNm8/d+QdDMdzYUaHrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cef23a6a16c930d63d5abd88bf6d56bdbfc457628d8a17398056f73b3f43cb3b","last_reissued_at":"2026-05-18T04:40:19.261840Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:19.261840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.5076","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":"2026-05-18T04:40:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z8ubV9iusk5JNVPc+KtrF2M9EHIRvuYFlXG/FMOAMzFImys23RjMsi8PbRmq7lwCbgZJOAvh0fG9u6oqtk3wCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T19:37:12.780500Z"},"content_sha256":"20cb19514b1cc468d6eb0e91e079f6bf751ebe282d7412440b82b38c72c12117","schema_version":"1.0","event_id":"sha256:20cb19514b1cc468d6eb0e91e079f6bf751ebe282d7412440b82b38c72c12117"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:Z3ZDU2QWZEYNMPK2XWEL63KWXW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Arnold's and Kazhdan's equidistribution problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexander Gorodnik, Amos Nevo","submitted_at":"2010-09-26T09:13:31Z","abstract_excerpt":"We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite, which implies the existence of an associated limiting distribution, possibly different than the invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5076","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":"2026-05-18T04:40:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VqZDxbzfaALrhiqKzmWUJsqgTvbsccGfGKJME0aEMnmr0fMlkZ6Xr4YGqun7uDBqWw4+Q089Q9/NfRJQ45IYCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T19:37:12.780848Z"},"content_sha256":"e06bb0dc0b5fb75c6ceac88c9a2d04c8a1b93746609917235de878f3855c0d11","schema_version":"1.0","event_id":"sha256:e06bb0dc0b5fb75c6ceac88c9a2d04c8a1b93746609917235de878f3855c0d11"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW/bundle.json","state_url":"https://pith.science/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW/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":"2026-06-23T19:37:12Z","links":{"resolver":"https://pith.science/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW","bundle":"https://pith.science/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW/bundle.json","state":"https://pith.science/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z3ZDU2QWZEYNMPK2XWEL63KWXW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:Z3ZDU2QWZEYNMPK2XWEL63KWXW","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":"b8464f40d49e64722f29b7306ef4664d02be8d2b6565fe60ed27dd566a9ab85c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-26T09:13:31Z","title_canon_sha256":"591a9a1e80aa4f3499e276eca154696b403c82f53634c12d48737a11fd72bc9f"},"schema_version":"1.0","source":{"id":"1009.5076","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.5076","created_at":"2026-05-18T04:40:19Z"},{"alias_kind":"arxiv_version","alias_value":"1009.5076v1","created_at":"2026-05-18T04:40:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5076","created_at":"2026-05-18T04:40:19Z"},{"alias_kind":"pith_short_12","alias_value":"Z3ZDU2QWZEYN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Z3ZDU2QWZEYNMPK2","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Z3ZDU2QW","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:e06bb0dc0b5fb75c6ceac88c9a2d04c8a1b93746609917235de878f3855c0d11","target":"graph","created_at":"2026-05-18T04:40:19Z","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 consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite, which implies the existence of an associated limiting distribution, possibly different than the invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, wh","authors_text":"Alexander Gorodnik, Amos Nevo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-26T09:13:31Z","title":"On Arnold's and Kazhdan's equidistribution problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5076","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":"sha256:20cb19514b1cc468d6eb0e91e079f6bf751ebe282d7412440b82b38c72c12117","target":"record","created_at":"2026-05-18T04:40:19Z","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":"b8464f40d49e64722f29b7306ef4664d02be8d2b6565fe60ed27dd566a9ab85c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-26T09:13:31Z","title_canon_sha256":"591a9a1e80aa4f3499e276eca154696b403c82f53634c12d48737a11fd72bc9f"},"schema_version":"1.0","source":{"id":"1009.5076","kind":"arxiv","version":1}},"canonical_sha256":"cef23a6a16c930d63d5abd88bf6d56bdbfc457628d8a17398056f73b3f43cb3b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cef23a6a16c930d63d5abd88bf6d56bdbfc457628d8a17398056f73b3f43cb3b","first_computed_at":"2026-05-18T04:40:19.261840Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:19.261840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N5G3JnklhQ1Mhf4/n5XgaIRSP0atnJOWFHP9B68CJTIa/HAikoCP79j4Ni7WC/z03txNm8/d+QdDMdzYUaHrCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:19.262488Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.5076","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20cb19514b1cc468d6eb0e91e079f6bf751ebe282d7412440b82b38c72c12117","sha256:e06bb0dc0b5fb75c6ceac88c9a2d04c8a1b93746609917235de878f3855c0d11"],"state_sha256":"a1f446763d0a818ecde8abf10c84ec5abced39e1bced8fedb78d8cf3501f3a51"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zibzi8AAXQUZiMq2noUWmGMzbW9fRn6Vk2aetmICbPFOLeUxSCmuRicuxH6fVTRbkWqyiin/hzVai3q0mflvCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T19:37:12.782813Z","bundle_sha256":"3dabcd151db8e76f1f90f7e3025cce9dbed69b3e7595b46db7533bb85b6693b6"}}