{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:7T6HH4K3PTMUO5D3GMUCDVUP4J","short_pith_number":"pith:7T6HH4K3","schema_version":"1.0","canonical_sha256":"fcfc73f15b7cd947747b332821d68fe25ac806af4ff4b9e6b84ca33e5b96b12e","source":{"kind":"arxiv","id":"1503.02203","version":1},"attestation_state":"computed","paper":{"title":"Variations on Dirichlet's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Simmons, Lior Fishman","submitted_at":"2015-03-07T19:30:20Z","abstract_excerpt":"We give a necessary and sufficient condition for the following property of an integer $d\\in\\mathbb N$ and a pair $(a,A)\\in\\mathbb R^2$: There exist $\\kappa > 0$ and $Q_0\\in\\mathbb N$ such that for all $\\mathbf x\\in \\mathbb R^d$ and $Q\\geq Q_0$, there exists $\\mathbf p/q\\in\\mathbb Q^d$ such that $1\\leq q\\leq Q$ and $\\|\\mathbf x - \\mathbf p/q\\| \\leq \\kappa q^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $\\kappa = Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing t"},"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":"1503.02203","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-07T19:30:20Z","cross_cats_sorted":[],"title_canon_sha256":"8e98c0ddd3ad7a4c2c1decf1560cf264a76c0add4a19fde3cc2e7005712df9fa","abstract_canon_sha256":"b8ee16e53fcc59a0a782e56f592724a131b0811c615e82016e3293889f626562"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:25.630777Z","signature_b64":"KNELSFWmvjqA40owcvN6p70Zf0qMP+q7eVotWLn/bBccfUNdrGQnXQuY60wVCDchPPvxAzzF4fTc8O13CkJcBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcfc73f15b7cd947747b332821d68fe25ac806af4ff4b9e6b84ca33e5b96b12e","last_reissued_at":"2026-05-18T02:25:25.630343Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:25.630343Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Variations on Dirichlet's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Simmons, Lior Fishman","submitted_at":"2015-03-07T19:30:20Z","abstract_excerpt":"We give a necessary and sufficient condition for the following property of an integer $d\\in\\mathbb N$ and a pair $(a,A)\\in\\mathbb R^2$: There exist $\\kappa > 0$ and $Q_0\\in\\mathbb N$ such that for all $\\mathbf x\\in \\mathbb R^d$ and $Q\\geq Q_0$, there exists $\\mathbf p/q\\in\\mathbb Q^d$ such that $1\\leq q\\leq Q$ and $\\|\\mathbf x - \\mathbf p/q\\| \\leq \\kappa q^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $\\kappa = Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02203","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1503.02203","created_at":"2026-05-18T02:25:25.630409+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.02203v1","created_at":"2026-05-18T02:25:25.630409+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.02203","created_at":"2026-05-18T02:25:25.630409+00:00"},{"alias_kind":"pith_short_12","alias_value":"7T6HH4K3PTMU","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7T6HH4K3PTMUO5D3","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7T6HH4K3","created_at":"2026-05-18T12:29:10.953037+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/7T6HH4K3PTMUO5D3GMUCDVUP4J","json":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J.json","graph_json":"https://pith.science/api/pith-number/7T6HH4K3PTMUO5D3GMUCDVUP4J/graph.json","events_json":"https://pith.science/api/pith-number/7T6HH4K3PTMUO5D3GMUCDVUP4J/events.json","paper":"https://pith.science/paper/7T6HH4K3"},"agent_actions":{"view_html":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J","download_json":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J.json","view_paper":"https://pith.science/paper/7T6HH4K3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.02203&json=true","fetch_graph":"https://pith.science/api/pith-number/7T6HH4K3PTMUO5D3GMUCDVUP4J/graph.json","fetch_events":"https://pith.science/api/pith-number/7T6HH4K3PTMUO5D3GMUCDVUP4J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J/action/storage_attestation","attest_author":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J/action/author_attestation","sign_citation":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J/action/citation_signature","submit_replication":"https://pith.science/pith/7T6HH4K3PTMUO5D3GMUCDVUP4J/action/replication_record"}},"created_at":"2026-05-18T02:25:25.630409+00:00","updated_at":"2026-05-18T02:25:25.630409+00:00"}