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Those characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any tw"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0908.3687","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-08-25T20:57:57Z","cross_cats_sorted":["math.GR","math.GT"],"title_canon_sha256":"5437f5d40662238b2011c4e9bab58ca0a36e6768875e156064fdefd20627761d","abstract_canon_sha256":"5352e9a2599aaf735a890ed8c58b20142ae2080d15aade671161e73ad01b18b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:29.101497Z","signature_b64":"dyDTBO+hi4D6lKGeo0L6hI78ouZ77f28/Oi2ZlEuvoSc93hzi23XGvws47Ey/nFm7V5DD4F8M//gtuTQc+bXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e4e498cee2de07bf7a152a13333c6fa35c8bc38598ff5f5f110e62076414f2a","last_reissued_at":"2026-05-18T04:11:29.100694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:29.100694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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