{"paper":{"title":"Interiors of continuous images of the middle-third Cantor set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Kan Jiang, Lifeng Xi","submitted_at":"2018-09-06T08:42:48Z","abstract_excerpt":"Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\\subset \\mathbb{R}^{2}$. Denote the image \\begin{equation*} f_{U}(C,C)=\\{f(x,y):(x,y)\\in (C\\times C)\\cap U\\}. \\end{equation*} If $\\partial _{x}f$, $\\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\\in (C\\times C)\\cap U$ such that \\begin{equation*} 1<\\left\\vert \\frac{\\partial _{x}f|_{(x_{0},y_{0})}}{\\partial _{y}f|_{(x_{0},y_{0})}}\\right\\vert <3\\text{ or }1<\\left\\vert \\frac{\\partial _{y}f|_{(x_{0},y_{0})}}{\\partial _{x}f|_{(x_{0},y_{0})}}\\right\\vert <3, \\end{equation*} then $f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.01880","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"}