{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:EJ4ILH3URKAXWSLZ7UM5GXCVKL","short_pith_number":"pith:EJ4ILH3U","canonical_record":{"source":{"id":"1012.0675","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-03T09:34:15Z","cross_cats_sorted":[],"title_canon_sha256":"9b2a86010b2b5451f6da7fbd10e8a0d3b22c13a1346a94d7856b4acced4915e3","abstract_canon_sha256":"a14c8550c6f43b8a12954afcbeb290a7773c01697ee9ea1f62cf216c755ed116"},"schema_version":"1.0"},"canonical_sha256":"2278859f748a817b4979fd19d35c5552de9517eb825dc412c9a18bebd45c88d9","source":{"kind":"arxiv","id":"1012.0675","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0675","created_at":"2026-05-18T03:13:43Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0675v2","created_at":"2026-05-18T03:13:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0675","created_at":"2026-05-18T03:13:43Z"},{"alias_kind":"pith_short_12","alias_value":"EJ4ILH3URKAX","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EJ4ILH3URKAXWSLZ","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EJ4ILH3U","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:EJ4ILH3URKAXWSLZ7UM5GXCVKL","target":"record","payload":{"canonical_record":{"source":{"id":"1012.0675","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-03T09:34:15Z","cross_cats_sorted":[],"title_canon_sha256":"9b2a86010b2b5451f6da7fbd10e8a0d3b22c13a1346a94d7856b4acced4915e3","abstract_canon_sha256":"a14c8550c6f43b8a12954afcbeb290a7773c01697ee9ea1f62cf216c755ed116"},"schema_version":"1.0"},"canonical_sha256":"2278859f748a817b4979fd19d35c5552de9517eb825dc412c9a18bebd45c88d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:43.847794Z","signature_b64":"FuW2ty5M4CKcm3F1A55sPYlsTWY+zZmiXsrrwAPVJAMV9/FIK2Vju5JwdeVyRL0TdF5JMA2bfRrv06zNY4ILBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2278859f748a817b4979fd19d35c5552de9517eb825dc412c9a18bebd45c88d9","last_reissued_at":"2026-05-18T03:13:43.847171Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:43.847171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.0675","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":"2026-05-18T03:13:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3DezKk6LrKiWhvt7Xcw61WiGd6GK8FldP1sgNRX2Ppxow6jvwKkptc8AGTaTILttufgT4Uk12wNdUq6t3IZtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:12:31.295563Z"},"content_sha256":"5a10996d94555aafa258949dfe245977264256a932f2834787fa5f84bfa41a4e","schema_version":"1.0","event_id":"sha256:5a10996d94555aafa258949dfe245977264256a932f2834787fa5f84bfa41a4e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:EJ4ILH3URKAXWSLZ7UM5GXCVKL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiplicative zero-one laws and metric number theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alan Haynes, Sanju Velani, Victor Beresnevich","submitted_at":"2010-12-03T09:34:15Z","abstract_excerpt":"We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0675","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":"2026-05-18T03:13:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/epz6epyae/8hS34sIWn4JHhot7dINyCRly0MCww+ZHRa/xf6AxMykBdWxwgxB5z1E3tMD/akpWB22G7MXhZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:12:31.296234Z"},"content_sha256":"7610787b4f78dfdd95d4aad2895e69b4843f450914b1b2f38071f9c564042e19","schema_version":"1.0","event_id":"sha256:7610787b4f78dfdd95d4aad2895e69b4843f450914b1b2f38071f9c564042e19"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL/bundle.json","state_url":"https://pith.science/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL/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":"2026-06-08T19:12:31Z","links":{"resolver":"https://pith.science/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL","bundle":"https://pith.science/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL/bundle.json","state":"https://pith.science/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EJ4ILH3URKAXWSLZ7UM5GXCVKL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EJ4ILH3URKAXWSLZ7UM5GXCVKL","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":"a14c8550c6f43b8a12954afcbeb290a7773c01697ee9ea1f62cf216c755ed116","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-03T09:34:15Z","title_canon_sha256":"9b2a86010b2b5451f6da7fbd10e8a0d3b22c13a1346a94d7856b4acced4915e3"},"schema_version":"1.0","source":{"id":"1012.0675","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0675","created_at":"2026-05-18T03:13:43Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0675v2","created_at":"2026-05-18T03:13:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0675","created_at":"2026-05-18T03:13:43Z"},{"alias_kind":"pith_short_12","alias_value":"EJ4ILH3URKAX","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EJ4ILH3URKAXWSLZ","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EJ4ILH3U","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:7610787b4f78dfdd95d4aad2895e69b4843f450914b1b2f38071f9c564042e19","target":"graph","created_at":"2026-05-18T03:13:43Z","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 develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions.","authors_text":"Alan Haynes, Sanju Velani, Victor Beresnevich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-03T09:34:15Z","title":"Multiplicative zero-one laws and metric number theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0675","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":"sha256:5a10996d94555aafa258949dfe245977264256a932f2834787fa5f84bfa41a4e","target":"record","created_at":"2026-05-18T03:13:43Z","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":"a14c8550c6f43b8a12954afcbeb290a7773c01697ee9ea1f62cf216c755ed116","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-03T09:34:15Z","title_canon_sha256":"9b2a86010b2b5451f6da7fbd10e8a0d3b22c13a1346a94d7856b4acced4915e3"},"schema_version":"1.0","source":{"id":"1012.0675","kind":"arxiv","version":2}},"canonical_sha256":"2278859f748a817b4979fd19d35c5552de9517eb825dc412c9a18bebd45c88d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2278859f748a817b4979fd19d35c5552de9517eb825dc412c9a18bebd45c88d9","first_computed_at":"2026-05-18T03:13:43.847171Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:43.847171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FuW2ty5M4CKcm3F1A55sPYlsTWY+zZmiXsrrwAPVJAMV9/FIK2Vju5JwdeVyRL0TdF5JMA2bfRrv06zNY4ILBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:43.847794Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.0675","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a10996d94555aafa258949dfe245977264256a932f2834787fa5f84bfa41a4e","sha256:7610787b4f78dfdd95d4aad2895e69b4843f450914b1b2f38071f9c564042e19"],"state_sha256":"d8eace5fc599d4d899a1869ebc26be88cc90aa9a0ba076092895c88f4981491a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mQhAMur43RmK1vEbQQ8aAQ680CoJnaOqIaJHgY4VO6E6p9xbaFN1jc4e4jPHFPNW5NbGMyLSMDChjyoR1ZXmCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T19:12:31.299872Z","bundle_sha256":"ce1a32a75d1e2880a579b2001cdf3e08a81ead214d27f387dda2d5e80090b373"}}