{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:P3YTFIDGRABYSYZJZO5AACFMBB","short_pith_number":"pith:P3YTFIDG","schema_version":"1.0","canonical_sha256":"7ef132a0668803896329cbba0008ac08767762b51928c0f0f624fd7139094ea0","source":{"kind":"arxiv","id":"1208.4786","version":2},"attestation_state":"computed","paper":{"title":"Schanuel's theorem for heights defined via extension fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christopher Frei, Martin Widmer","submitted_at":"2012-08-23T15:49:51Z","abstract_excerpt":"Let $k$ be a number field, let $\\theta$ be a nonzero algebraic number, and let $H(\\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of $\\alpha \\in k$ with $H(\\alpha \\theta)\\leq X$.\n  We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof.\n  Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's probl"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1208.4786","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-08-23T15:49:51Z","cross_cats_sorted":[],"title_canon_sha256":"63e9fc9a25b8a6236fce7bc1dfb78c682482f5fbc5d14673ff036abf27663d22","abstract_canon_sha256":"a3465ae74fe85f579a9ee38fdee669edb03f3f2693d4e0e72d6162adf9895cc9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:42.326407Z","signature_b64":"Exp3rubYtnixESjkSaQC08nS5BDzsoM9R1e7mj7p3c1WjFqCLhEgeuRO2UpQmyROmXIIECca3G5GcLVZBip4DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ef132a0668803896329cbba0008ac08767762b51928c0f0f624fd7139094ea0","last_reissued_at":"2026-05-18T02:52:42.325770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:42.325770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Schanuel's theorem for heights defined via extension fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christopher Frei, Martin Widmer","submitted_at":"2012-08-23T15:49:51Z","abstract_excerpt":"Let $k$ be a number field, let $\\theta$ be a nonzero algebraic number, and let $H(\\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of $\\alpha \\in k$ with $H(\\alpha \\theta)\\leq X$.\n  We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof.\n  Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's probl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4786","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1208.4786","created_at":"2026-05-18T02:52:42.325869+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.4786v2","created_at":"2026-05-18T02:52:42.325869+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4786","created_at":"2026-05-18T02:52:42.325869+00:00"},{"alias_kind":"pith_short_12","alias_value":"P3YTFIDGRABY","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"P3YTFIDGRABYSYZJ","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"P3YTFIDG","created_at":"2026-05-18T12:27:18.751474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB","json":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB.json","graph_json":"https://pith.science/api/pith-number/P3YTFIDGRABYSYZJZO5AACFMBB/graph.json","events_json":"https://pith.science/api/pith-number/P3YTFIDGRABYSYZJZO5AACFMBB/events.json","paper":"https://pith.science/paper/P3YTFIDG"},"agent_actions":{"view_html":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB","download_json":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB.json","view_paper":"https://pith.science/paper/P3YTFIDG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.4786&json=true","fetch_graph":"https://pith.science/api/pith-number/P3YTFIDGRABYSYZJZO5AACFMBB/graph.json","fetch_events":"https://pith.science/api/pith-number/P3YTFIDGRABYSYZJZO5AACFMBB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB/action/storage_attestation","attest_author":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB/action/author_attestation","sign_citation":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB/action/citation_signature","submit_replication":"https://pith.science/pith/P3YTFIDGRABYSYZJZO5AACFMBB/action/replication_record"}},"created_at":"2026-05-18T02:52:42.325869+00:00","updated_at":"2026-05-18T02:52:42.325869+00:00"}