{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:EKGBSO5OJX7ESQSX4CIPFKKKQW","short_pith_number":"pith:EKGBSO5O","canonical_record":{"source":{"id":"2204.07357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-15T07:34:45Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"c793bcc994806ae6ec9c3a7a3fb23beb556b4f81fcb4d4ab8adc6f95cfe51c53","abstract_canon_sha256":"70ecd2b2930eb924b72a54d32d69308ebeb9133acaa81a8c0ceb3f7ad4576263"},"schema_version":"1.0"},"canonical_sha256":"228c193bae4dfe494257e090f2a94a858d41ea56c37fb44d64df104f0a313f5b","source":{"kind":"arxiv","id":"2204.07357","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2204.07357","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"arxiv_version","alias_value":"2204.07357v1","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.07357","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"pith_short_12","alias_value":"EKGBSO5OJX7E","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"pith_short_16","alias_value":"EKGBSO5OJX7ESQSX","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"pith_short_8","alias_value":"EKGBSO5O","created_at":"2026-07-05T04:15:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:EKGBSO5OJX7ESQSX4CIPFKKKQW","target":"record","payload":{"canonical_record":{"source":{"id":"2204.07357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-15T07:34:45Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"c793bcc994806ae6ec9c3a7a3fb23beb556b4f81fcb4d4ab8adc6f95cfe51c53","abstract_canon_sha256":"70ecd2b2930eb924b72a54d32d69308ebeb9133acaa81a8c0ceb3f7ad4576263"},"schema_version":"1.0"},"canonical_sha256":"228c193bae4dfe494257e090f2a94a858d41ea56c37fb44d64df104f0a313f5b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:15:03.107588Z","signature_b64":"o71sQ+mABEGWMkMZobA30qePmh+FbVMNJ2Qcz0ItXT0F/vm5yYXIZXA9/W/NNGn/0fchpHLwAnFe0TnlosfsCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"228c193bae4dfe494257e090f2a94a858d41ea56c37fb44d64df104f0a313f5b","last_reissued_at":"2026-07-05T04:15:03.107163Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:15:03.107163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2204.07357","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":"2026-07-05T04:15:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OS9bw6Lz1CWDuo1l6xR4p3EOxYQ/5MM+4/dltbD95D14dlzENphwkTrpL6soXRrWIJqrGqZKSvOFrNLgRisDAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T07:04:02.417194Z"},"content_sha256":"b49f7cd7ceee82191105a4c7e222f6edaeb27eb466e41b18534fd852f0abfa28","schema_version":"1.0","event_id":"sha256:b49f7cd7ceee82191105a4c7e222f6edaeb27eb466e41b18534fd852f0abfa28"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:EKGBSO5OJX7ESQSX4CIPFKKKQW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rational numbers in $\\times b$-invariant sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Bing Li, Ruofan Li, Yufeng Wu","submitted_at":"2022-04-15T07:34:45Z","abstract_excerpt":"Let $b \\geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \\subset [0,1)$ such that $A \\cap \\mathbb{Q}$ is invariant under $\\times b$ operation, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.07357","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2204.07357/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T04:15:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+DADzaD3DXlsQ/HnxX6QZ5BY2KGFGqk2HPh05Pk5ZlE9xEtjsaT+rdBxyg81rPoYuqKuKi9bkEuCz7jkHPOWAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T07:04:02.417581Z"},"content_sha256":"d2aa79576ee48e6fd21f133af0abc9ca4dd42afabe1188b0922b3247a30478b5","schema_version":"1.0","event_id":"sha256:d2aa79576ee48e6fd21f133af0abc9ca4dd42afabe1188b0922b3247a30478b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW/bundle.json","state_url":"https://pith.science/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW/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":"2026-07-06T07:04:02Z","links":{"resolver":"https://pith.science/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW","bundle":"https://pith.science/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW/bundle.json","state":"https://pith.science/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EKGBSO5OJX7ESQSX4CIPFKKKQW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:EKGBSO5OJX7ESQSX4CIPFKKKQW","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":"70ecd2b2930eb924b72a54d32d69308ebeb9133acaa81a8c0ceb3f7ad4576263","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-15T07:34:45Z","title_canon_sha256":"c793bcc994806ae6ec9c3a7a3fb23beb556b4f81fcb4d4ab8adc6f95cfe51c53"},"schema_version":"1.0","source":{"id":"2204.07357","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2204.07357","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"arxiv_version","alias_value":"2204.07357v1","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.07357","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"pith_short_12","alias_value":"EKGBSO5OJX7E","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"pith_short_16","alias_value":"EKGBSO5OJX7ESQSX","created_at":"2026-07-05T04:15:03Z"},{"alias_kind":"pith_short_8","alias_value":"EKGBSO5O","created_at":"2026-07-05T04:15:03Z"}],"graph_snapshots":[{"event_id":"sha256:d2aa79576ee48e6fd21f133af0abc9ca4dd42afabe1188b0922b3247a30478b5","target":"graph","created_at":"2026-07-05T04:15:03Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2204.07357/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $b \\geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \\subset [0,1)$ such that $A \\cap \\mathbb{Q}$ is invariant under $\\times b$ operation, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained.","authors_text":"Bing Li, Ruofan Li, Yufeng Wu","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-15T07:34:45Z","title":"Rational numbers in $\\times b$-invariant sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.07357","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:b49f7cd7ceee82191105a4c7e222f6edaeb27eb466e41b18534fd852f0abfa28","target":"record","created_at":"2026-07-05T04:15:03Z","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":"70ecd2b2930eb924b72a54d32d69308ebeb9133acaa81a8c0ceb3f7ad4576263","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-04-15T07:34:45Z","title_canon_sha256":"c793bcc994806ae6ec9c3a7a3fb23beb556b4f81fcb4d4ab8adc6f95cfe51c53"},"schema_version":"1.0","source":{"id":"2204.07357","kind":"arxiv","version":1}},"canonical_sha256":"228c193bae4dfe494257e090f2a94a858d41ea56c37fb44d64df104f0a313f5b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"228c193bae4dfe494257e090f2a94a858d41ea56c37fb44d64df104f0a313f5b","first_computed_at":"2026-07-05T04:15:03.107163Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:15:03.107163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o71sQ+mABEGWMkMZobA30qePmh+FbVMNJ2Qcz0ItXT0F/vm5yYXIZXA9/W/NNGn/0fchpHLwAnFe0TnlosfsCA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:15:03.107588Z","signed_message":"canonical_sha256_bytes"},"source_id":"2204.07357","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b49f7cd7ceee82191105a4c7e222f6edaeb27eb466e41b18534fd852f0abfa28","sha256:d2aa79576ee48e6fd21f133af0abc9ca4dd42afabe1188b0922b3247a30478b5"],"state_sha256":"06db9a49c539a142f08b286912358c31cc759c5e2ce776d9ebdb9beb396e2e52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xoNG0l3VwSdU8Q0eeDN0sG0r63RgoipsCVrQMiMzDKoMPExYsxLrhBzIn9cIR8fsU05Fo7gSSuK2pkx9HvUVAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T07:04:02.420563Z","bundle_sha256":"ee0eddda58a45421f0ea52af02a958e5a8bff2e28a94f441b4f7169c9e78d364"}}