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Assume that $\\mathcal N$ is a $\\delta$-net in the unit ball of $X$ and that $\\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem.\n  We also obtain"},"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":"1110.5368","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-24T22:32:43Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"5b507dd3ff8d33ab2440195af57d85b30fb35c6d81592cb6da39b29407f65747","abstract_canon_sha256":"239e904e267fe061f5f44e4bf829fea906e1c57e57da839dbd06631dca3104f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:19.130043Z","signature_b64":"97xAz0Gmax73IvZdoiAZmFsY1YX4wDnWa33qM9K1r8fLCwRaHeWmoS9aVt8fR+JQbGTXOINf5DPygzRKtsr+Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d6679f5fa2e12f8866d7378f30cafadf4b6b43d093cb122952a3af1c30100f8","last_reissued_at":"2026-05-18T02:26:19.129481Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:19.129481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bourgain's discretization theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Assaf Naor, Gideon Schechtman, Ohad Giladi","submitted_at":"2011-10-24T22:32:43Z","abstract_excerpt":"Bourgain's discretization theorem asserts that there exists a universal constant $C\\in (0,\\infty)$ with the following property. 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