{"paper":{"title":"Affine embeddings of Cantor sets on the line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Amir Algom","submitted_at":"2016-07-11T08:04:36Z","abstract_excerpt":"Let $s\\in (0,1)$, and let $F\\subset \\mathbb{R}$ be a self similar set such that $0 < \\dim_H F \\leq s$ . We prove that there exists $\\delta= \\delta(s) >0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$ and $0 \\leq \\dim_H E - \\dim_H F < \\delta$ then (under some mild conditions on $E$ and $F$) the contraction ratios of $E$ and $F$ are logarithmically commensurable. This provides more evidence for a Conjecture of Feng, Huang, and Rao, that states that these contraction ratios are logarithmically commensurable whenever $F$ admits an affine embedding into $E$ (un"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02849","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"}