{"paper":{"title":"Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AP","authors_text":"Terence Tao","submitted_at":"2018-11-22T16:11:36Z","abstract_excerpt":"Let $H := \\begin{pmatrix} 1 & {\\mathbf R} & {\\mathbf R} \\\\ 0 & 1 &{\\mathbf R} \\\\ 0 & 0 & 1 \\end{pmatrix}$ denote the Heisenberg group with the usual Carnot-Carath\\'eodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipchitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \\varepsilon < 1$, the snowflaked metric space $(H,d^{1-\\varepsilon})$ embeds into an infinite-dimensional Hilbert space with distortion $O( \\varepsilon^{-1/2} )$. This distortion bound was shown by Austin, Naor, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.09223","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"}