{"paper":{"title":"Qusisymmetric dimension distortion of Ahlfors regular subsets of a metric space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CV","authors_text":"Christopher J. Bishop, Hrant Hakobyan, Marshall Williams","submitted_at":"2012-11-01T17:23:35Z","abstract_excerpt":"We show that if $f:X\\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\\dim_H f(E)\\leq\\dim_H E$ for \"almost every\" bounded Ahlfors regular set $E\\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then $\\dim_H f(E)=\\dim_H E$ for \"almost every\" Ahlfors regular set $E\\subset X$. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if $f$ is a quasiconformal map of $\\mathbb{R}^N$, $N\\geq 2$, then for Lebesgue a.e. $y\\in\\mathbb{R}^N$ we have $\\dim_H f(y+E) = \\dim_H E$. A si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.0233","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"}