{"paper":{"title":"Affine embeddings of Cantor sets in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Amir Algom","submitted_at":"2017-09-12T15:18:43Z","abstract_excerpt":"Let $F,E\\subseteq \\mathbb{R}^2$ be two self similar sets. First, assuming $F$ is generated by an IFS $\\Phi$ with strong separation, we characterize the affine maps $g:\\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$ such that $g(F)\\subseteq F$. Our analysis depends on the cardinality of the group $G_\\Phi$ generated by the orthogonal parts of the similarities in $\\Phi$. When $|G_\\Phi|=\\infty$ we show that any such self embedding must be a similarity, and so (by the results of Elekes, Keleti and M\\'ath\\'{e}) some power of its orthogonal part lies in $G_\\Phi$. When $|G_\\Phi| < \\infty$ and $\\Phi$ has a unif"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.03906","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"}