{"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"}