{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7PFLA3YTU7E2GRT6PFYC2JUYSD","short_pith_number":"pith:7PFLA3YT","canonical_record":{"source":{"id":"1212.3525","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-14T16:48:27Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"833ee00544bebf6135b0adbb012af32a52bcb0a90cdd1c9641a8c90ac616a390","abstract_canon_sha256":"486de4ccfc106823797e7670d0d1dda86cee43c3c56eb1d0d1f77707446b3276"},"schema_version":"1.0"},"canonical_sha256":"fbcab06f13a7c9a3467e79702d269890f0493c3f91ee13a9b86a0b2087d02490","source":{"kind":"arxiv","id":"1212.3525","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.3525","created_at":"2026-05-18T03:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"1212.3525v1","created_at":"2026-05-18T03:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.3525","created_at":"2026-05-18T03:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"7PFLA3YTU7E2","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7PFLA3YTU7E2GRT6","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7PFLA3YT","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7PFLA3YTU7E2GRT6PFYC2JUYSD","target":"record","payload":{"canonical_record":{"source":{"id":"1212.3525","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-14T16:48:27Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"833ee00544bebf6135b0adbb012af32a52bcb0a90cdd1c9641a8c90ac616a390","abstract_canon_sha256":"486de4ccfc106823797e7670d0d1dda86cee43c3c56eb1d0d1f77707446b3276"},"schema_version":"1.0"},"canonical_sha256":"fbcab06f13a7c9a3467e79702d269890f0493c3f91ee13a9b86a0b2087d02490","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:24.399351Z","signature_b64":"MnaO9hGjhbYcTLpLjsQGjwSIJWtXCKGuBiU6Q9n53foDnDTDHh/43aiTctZz4nSN13XxCc+AcAd7XUQDZ+k2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbcab06f13a7c9a3467e79702d269890f0493c3f91ee13a9b86a0b2087d02490","last_reissued_at":"2026-05-18T03:38:24.398683Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:24.398683Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.3525","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":"2026-05-18T03:38:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BxgVB6dZhxgBXpECFVvdIwoPqU4o8Ohxax7MSwK3UiigUXaxlFLgdERybtWssu8ovCH3KquqQKjrPBQwlrTIDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T20:00:34.057900Z"},"content_sha256":"a65373adef5f9da5b548b263815e3748fffbc51783aa9edc943f26541cde6515","schema_version":"1.0","event_id":"sha256:a65373adef5f9da5b548b263815e3748fffbc51783aa9edc943f26541cde6515"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7PFLA3YTU7E2GRT6PFYC2JUYSD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Notes on Thin Matrix Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Peter Sarnak","submitted_at":"2012-12-14T16:48:27Z","abstract_excerpt":"These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3525","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":"2026-05-18T03:38:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eJmIvX2aFWq9NUWHcGHg7xgrhPWPEbRdqLotTyAqSp+sypq36Q6RP/BJe57NyHwcejK0IpOM1zxisFXLSQSjCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T20:00:34.058732Z"},"content_sha256":"e8d17991490918bcdc466638097e100b8e92d035849e4970579a1056e9c9286e","schema_version":"1.0","event_id":"sha256:e8d17991490918bcdc466638097e100b8e92d035849e4970579a1056e9c9286e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD/bundle.json","state_url":"https://pith.science/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD/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":"2026-08-06T20:00:34Z","links":{"resolver":"https://pith.science/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD","bundle":"https://pith.science/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD/bundle.json","state":"https://pith.science/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7PFLA3YTU7E2GRT6PFYC2JUYSD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7PFLA3YTU7E2GRT6PFYC2JUYSD","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":"486de4ccfc106823797e7670d0d1dda86cee43c3c56eb1d0d1f77707446b3276","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-14T16:48:27Z","title_canon_sha256":"833ee00544bebf6135b0adbb012af32a52bcb0a90cdd1c9641a8c90ac616a390"},"schema_version":"1.0","source":{"id":"1212.3525","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.3525","created_at":"2026-05-18T03:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"1212.3525v1","created_at":"2026-05-18T03:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.3525","created_at":"2026-05-18T03:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"7PFLA3YTU7E2","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7PFLA3YTU7E2GRT6","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7PFLA3YT","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:e8d17991490918bcdc466638097e100b8e92d035849e4970579a1056e9c9286e","target":"graph","created_at":"2026-05-18T03:38:24Z","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":"These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.","authors_text":"Peter Sarnak","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-14T16:48:27Z","title":"Notes on Thin Matrix Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3525","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:a65373adef5f9da5b548b263815e3748fffbc51783aa9edc943f26541cde6515","target":"record","created_at":"2026-05-18T03:38:24Z","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":"486de4ccfc106823797e7670d0d1dda86cee43c3c56eb1d0d1f77707446b3276","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-14T16:48:27Z","title_canon_sha256":"833ee00544bebf6135b0adbb012af32a52bcb0a90cdd1c9641a8c90ac616a390"},"schema_version":"1.0","source":{"id":"1212.3525","kind":"arxiv","version":1}},"canonical_sha256":"fbcab06f13a7c9a3467e79702d269890f0493c3f91ee13a9b86a0b2087d02490","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbcab06f13a7c9a3467e79702d269890f0493c3f91ee13a9b86a0b2087d02490","first_computed_at":"2026-05-18T03:38:24.398683Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:24.398683Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MnaO9hGjhbYcTLpLjsQGjwSIJWtXCKGuBiU6Q9n53foDnDTDHh/43aiTctZz4nSN13XxCc+AcAd7XUQDZ+k2Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:24.399351Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.3525","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a65373adef5f9da5b548b263815e3748fffbc51783aa9edc943f26541cde6515","sha256:e8d17991490918bcdc466638097e100b8e92d035849e4970579a1056e9c9286e"],"state_sha256":"dd059a047f4720e1f9678865558ad18f5be54980761137bdf29b36bda59731a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wk+G0tmwoBoNAvUmMa9iLK+tQ1a5A/durSpM7yJdqhIEq+mNsL1f7K0M9bk0wTGar/z+W89b81Ldu+f1l6UgCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T20:00:34.065028Z","bundle_sha256":"2c9c564ffd0763ecb8a63425c777f03c8464774bb0fa177d5b4209380ecb1dc1"}}