{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:SCMLL7TFCBCARHOABWUNRLKEKV","short_pith_number":"pith:SCMLL7TF","schema_version":"1.0","canonical_sha256":"9098b5fe651044089dc00da8d8ad44555fdafd63e893f04eb94c05530d15e5cb","source":{"kind":"arxiv","id":"1609.06235","version":3},"attestation_state":"computed","paper":{"title":"Boundary convergence and path divergence sets for bounded analytic functions in the disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Trevor Richards","submitted_at":"2016-09-20T16:09:02Z","abstract_excerpt":"Let $f:\\mathbb{D}\\to\\mathbb{C}$ be a bounded analytic function. A set $K\\subset\\mathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $\\zeta$ as $z\\to1$ with $z\\in K$. $K$ is called a path divergence set for $f$ at $1$ if $f$ diverges along every path $\\gamma$ which lies in $K$ and approaches $1$. In this article, we show that for a path $\\gamma$ through the unit disk from $-1$ to $1$, if $f$ fails to converge along $\\gamma$, then either the region above $\\gamma$ or the region below $\\gamma$ is a path divergence set"},"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":"1609.06235","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-09-20T16:09:02Z","cross_cats_sorted":[],"title_canon_sha256":"503e79604f7954a2550952008f81d164a00a1f9cef199045a2405c1084753abe","abstract_canon_sha256":"08ba29d0d962e3271d15066ae55dfc1dbcfaf651f8fc820e8eaaf18b2e85b275"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:59.281148Z","signature_b64":"CDAw438iaBU47t8WD0qg3VNMR1iFjcoIqJlA01xp42tly0Vt9TXHcPjMvZKAuepxgs5+MDzRVLTUV83v/8AjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9098b5fe651044089dc00da8d8ad44555fdafd63e893f04eb94c05530d15e5cb","last_reissued_at":"2026-05-17T23:42:59.280663Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:59.280663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundary convergence and path divergence sets for bounded analytic functions in the disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Trevor Richards","submitted_at":"2016-09-20T16:09:02Z","abstract_excerpt":"Let $f:\\mathbb{D}\\to\\mathbb{C}$ be a bounded analytic function. A set $K\\subset\\mathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $\\zeta$ as $z\\to1$ with $z\\in K$. $K$ is called a path divergence set for $f$ at $1$ if $f$ diverges along every path $\\gamma$ which lies in $K$ and approaches $1$. In this article, we show that for a path $\\gamma$ through the unit disk from $-1$ to $1$, if $f$ fails to converge along $\\gamma$, then either the region above $\\gamma$ or the region below $\\gamma$ is a path divergence set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06235","kind":"arxiv","version":3},"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":"1609.06235","created_at":"2026-05-17T23:42:59.280755+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.06235v3","created_at":"2026-05-17T23:42:59.280755+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06235","created_at":"2026-05-17T23:42:59.280755+00:00"},{"alias_kind":"pith_short_12","alias_value":"SCMLL7TFCBCA","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"SCMLL7TFCBCARHOA","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"SCMLL7TF","created_at":"2026-05-18T12:30:44.179134+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/SCMLL7TFCBCARHOABWUNRLKEKV","json":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV.json","graph_json":"https://pith.science/api/pith-number/SCMLL7TFCBCARHOABWUNRLKEKV/graph.json","events_json":"https://pith.science/api/pith-number/SCMLL7TFCBCARHOABWUNRLKEKV/events.json","paper":"https://pith.science/paper/SCMLL7TF"},"agent_actions":{"view_html":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV","download_json":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV.json","view_paper":"https://pith.science/paper/SCMLL7TF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.06235&json=true","fetch_graph":"https://pith.science/api/pith-number/SCMLL7TFCBCARHOABWUNRLKEKV/graph.json","fetch_events":"https://pith.science/api/pith-number/SCMLL7TFCBCARHOABWUNRLKEKV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV/action/storage_attestation","attest_author":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV/action/author_attestation","sign_citation":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV/action/citation_signature","submit_replication":"https://pith.science/pith/SCMLL7TFCBCARHOABWUNRLKEKV/action/replication_record"}},"created_at":"2026-05-17T23:42:59.280755+00:00","updated_at":"2026-05-17T23:42:59.280755+00:00"}