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One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03,MatousekProblems07,ABN08,CGT10]), and is still open. We prove another result in this line of work:\n  The snowflake metric $d^{1/2}$ of a doubling set $S \\subset l_2$ embeds with constant distortion into $l_2^D$, for dimension $D$ that depend"},"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":"0907.5477","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2009-07-31T07:10:04Z","cross_cats_sorted":["cs.DS","math.FA","math.MG"],"title_canon_sha256":"25770348fd7b697923d67e9c671baa627cb00a2e79a03e3f318be42c05866a6b","abstract_canon_sha256":"c920371b4c9fc18df64322190fabd139377ee21b93a27d93ce922131aac02f92"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:55:58.605532Z","signature_b64":"48laXJCbQUEgGbYw+BHpEqlfehGKTlIMxOupEV04EgnbzmXYMCoYhzJhRlMAODgIHGJkTeObSCmEsTEkzE0xBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97e81efcca552c074efd093d45edc1a0a0c9f307fcb48ae46582d123411eac7d","last_reissued_at":"2026-05-18T01:55:58.605076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:55:58.605076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Nonlinear Approach to Dimension Reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.FA","math.MG"],"primary_cat":"cs.CG","authors_text":"Lee-Ad Gottlieb, Robert Krauthgamer","submitted_at":"2009-07-31T07:10:04Z","abstract_excerpt":"The $l_2$ flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. 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