{"paper":{"title":"Characterizing the Cantor bi-cube in asymptotic categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.MG","authors_text":"Ihor Zarichnyi, Taras Banakh","submitted_at":"2009-08-25T20:57:57Z","abstract_excerpt":"We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set $\\{\\sum_{i=-n}^\\infty\\frac{2x_i}{3^i}:n\\in\\IN ,\\;(x_i)_{i\\in\\IZ}\\in\\{0,1\\}^\\IZ\\}\\subset\\IR$, which is bi-uniformly equivalent to the Cantor bi-cube $2^{<\\IZ}=\\{(x_i)_{i\\in\\IZ}\\in \\{0,1\\}^\\IZ:\\exists n\\;\\forall i\\ge n\\;x_i=0\\}$ endowed with the metric $d((x_i),(y_i))=\\max_{i\\in\\IZ}2^i|x_i-y_i|$. Those characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any tw"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3687","kind":"arxiv","version":4},"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"}