{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ZGZZLV4LOOZVY7VPPFY6CCWA32","short_pith_number":"pith:ZGZZLV4L","canonical_record":{"source":{"id":"1705.00562","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-01T15:28:48Z","cross_cats_sorted":[],"title_canon_sha256":"376d4924d0d659cdec9e478b0c86ea44f7990e181d2f4aeb4d2ca45f28ce4899","abstract_canon_sha256":"9147ebfe8d168f16493bdb7247b370edfe2ac907759aa4498a998607477b2e6d"},"schema_version":"1.0"},"canonical_sha256":"c9b395d78b73b35c7eaf7971e10ac0dea5a06ea59ab131d2a96a2006b6276f1b","source":{"kind":"arxiv","id":"1705.00562","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.00562","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"arxiv_version","alias_value":"1705.00562v2","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.00562","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"pith_short_12","alias_value":"ZGZZLV4LOOZV","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZGZZLV4LOOZVY7VP","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZGZZLV4L","created_at":"2026-05-18T12:31:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ZGZZLV4LOOZVY7VPPFY6CCWA32","target":"record","payload":{"canonical_record":{"source":{"id":"1705.00562","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-01T15:28:48Z","cross_cats_sorted":[],"title_canon_sha256":"376d4924d0d659cdec9e478b0c86ea44f7990e181d2f4aeb4d2ca45f28ce4899","abstract_canon_sha256":"9147ebfe8d168f16493bdb7247b370edfe2ac907759aa4498a998607477b2e6d"},"schema_version":"1.0"},"canonical_sha256":"c9b395d78b73b35c7eaf7971e10ac0dea5a06ea59ab131d2a96a2006b6276f1b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:03.217627Z","signature_b64":"WmG1nInGguxVQsJS1+6bNzh2hg5Hmp42fLstfg0ESWCDmURA6d2zblmxkOclJwpDX6H+fQ26XQgfzT6YVhDEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9b395d78b73b35c7eaf7971e10ac0dea5a06ea59ab131d2a96a2006b6276f1b","last_reissued_at":"2026-05-18T00:16:03.217096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:03.217096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.00562","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":"2026-05-18T00:16:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zLF4qYHoMBi5O8OXZpn+9m28EnQKhiW1QqJGG2vU+OI12F0IhSly+8dO2Twx2wR5LLxqr10NHJHxU9pADOiLAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T19:29:42.347557Z"},"content_sha256":"6fea3bbaf035176501fc87caa2567e786ca86aad6394974d50189b9f164b9f93","schema_version":"1.0","event_id":"sha256:6fea3bbaf035176501fc87caa2567e786ca86aad6394974d50189b9f164b9f93"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ZGZZLV4LOOZVY7VPPFY6CCWA32","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Dirichlet approximation theorem for group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Clayton Petsche, Jeffrey D. Vaaler","submitted_at":"2017-05-01T15:28:48Z","abstract_excerpt":"If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.00562","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":"2026-05-18T00:16:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FkJk2SuDkRfllz2qZhtnw4DmYPAodZpiQNGThNTgNtHKaMJw+MVd5BZuMWrhzt2sfqO/Jv0NcXHw3Pqh8ERYAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T19:29:42.348225Z"},"content_sha256":"1fe595c941de7d4be796b83b1cdf5555bf41f8eec26f71e77b8e00dc1f40812f","schema_version":"1.0","event_id":"sha256:1fe595c941de7d4be796b83b1cdf5555bf41f8eec26f71e77b8e00dc1f40812f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32/bundle.json","state_url":"https://pith.science/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32/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":"2026-05-30T19:29:42Z","links":{"resolver":"https://pith.science/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32","bundle":"https://pith.science/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32/bundle.json","state":"https://pith.science/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZGZZLV4LOOZVY7VPPFY6CCWA32/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZGZZLV4LOOZVY7VPPFY6CCWA32","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":"9147ebfe8d168f16493bdb7247b370edfe2ac907759aa4498a998607477b2e6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-01T15:28:48Z","title_canon_sha256":"376d4924d0d659cdec9e478b0c86ea44f7990e181d2f4aeb4d2ca45f28ce4899"},"schema_version":"1.0","source":{"id":"1705.00562","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.00562","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"arxiv_version","alias_value":"1705.00562v2","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.00562","created_at":"2026-05-18T00:16:03Z"},{"alias_kind":"pith_short_12","alias_value":"ZGZZLV4LOOZV","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZGZZLV4LOOZVY7VP","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZGZZLV4L","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:1fe595c941de7d4be796b83b1cdf5555bf41f8eec26f71e77b8e00dc1f40812f","target":"graph","created_at":"2026-05-18T00:16:03Z","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":"If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting.","authors_text":"Clayton Petsche, Jeffrey D. Vaaler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-01T15:28:48Z","title":"A Dirichlet approximation theorem for group actions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.00562","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":"sha256:6fea3bbaf035176501fc87caa2567e786ca86aad6394974d50189b9f164b9f93","target":"record","created_at":"2026-05-18T00:16:03Z","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":"9147ebfe8d168f16493bdb7247b370edfe2ac907759aa4498a998607477b2e6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-01T15:28:48Z","title_canon_sha256":"376d4924d0d659cdec9e478b0c86ea44f7990e181d2f4aeb4d2ca45f28ce4899"},"schema_version":"1.0","source":{"id":"1705.00562","kind":"arxiv","version":2}},"canonical_sha256":"c9b395d78b73b35c7eaf7971e10ac0dea5a06ea59ab131d2a96a2006b6276f1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9b395d78b73b35c7eaf7971e10ac0dea5a06ea59ab131d2a96a2006b6276f1b","first_computed_at":"2026-05-18T00:16:03.217096Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:03.217096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WmG1nInGguxVQsJS1+6bNzh2hg5Hmp42fLstfg0ESWCDmURA6d2zblmxkOclJwpDX6H+fQ26XQgfzT6YVhDEBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:03.217627Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.00562","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6fea3bbaf035176501fc87caa2567e786ca86aad6394974d50189b9f164b9f93","sha256:1fe595c941de7d4be796b83b1cdf5555bf41f8eec26f71e77b8e00dc1f40812f"],"state_sha256":"0acda1089c33815838c700c136820b95a4fd84f29ebc9cbf92b2d1d1961761fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4oHZDtUN+2mwRNasU0pXrIouwVV3Hn5DHSJYC7T3PebdTCEMUAyncQXF8N13TwrlyPEbwi1dNYXxpWTpFrSHBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T19:29:42.352354Z","bundle_sha256":"3fa3528db4f7e7857084eca723ac0f2971173de9faaeddb86f90451719f55ded"}}