{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:BTDMM7VFUT55AMHOMSMOOTUKWU","short_pith_number":"pith:BTDMM7VF","schema_version":"1.0","canonical_sha256":"0cc6c67ea5a4fbd030ee6498e74e8ab503de5b101cda29560a0d5a60ebe33291","source":{"kind":"arxiv","id":"2607.11747","version":1},"attestation_state":"computed","paper":{"title":"Near-Optimal Embeddings of Constant-Dimensional Subspaces of $L^p$ into $\\ell_p^N$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Yi Li","submitted_at":"2026-07-13T16:06:14Z","abstract_excerpt":"For $d\\geq 2$, $p\\geq 1$ and $\\epsilon > 0$, let $N_{p}(d,\\epsilon)$ be the smallest integer $N$ such that every $d$-dimensional subspace of $L^{p}[0,1]$ admits a linear embedding into $\\ell_{p}^{N}$ with distortion at most $1+\\epsilon$. For fixed $d$ and $p$, the bound \\[ N_{p}(d,\\epsilon) = \\widetilde{O}_{d,p}\\!\\left(\\epsilon^{-2(d-1)/(d+2p)}\\right) \\] is established. For $p\\notin 2\\mathbb{Z}$, this is optimal up to logarithmic factors; for positive even integers $p$, isometric embeddings of dimension independent of $\\epsilon$ are known. The stated upper bound was previously known only for i"},"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":"2607.11747","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-13T16:06:14Z","cross_cats_sorted":[],"title_canon_sha256":"70ad87e74089059f07d2e3818ae8c62ecf178fcb2c4ebd43a1a6d007120f6971","abstract_canon_sha256":"09fb7a6a1fe3ede4330a69e1e89f23182c129578fd0cd3f60fdc68cc8c9c7874"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T02:22:22.701759Z","signature_b64":"Pvy7WzQTftEtKO0gFXkoxj3CcHCMXJCFR8MiSckoZ/JiO1QHJkEGvc/YxbLGiaAOKTmVlHx3ceYfYyQt3OyUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0cc6c67ea5a4fbd030ee6498e74e8ab503de5b101cda29560a0d5a60ebe33291","last_reissued_at":"2026-07-14T02:22:22.700871Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T02:22:22.700871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Near-Optimal Embeddings of Constant-Dimensional Subspaces of $L^p$ into $\\ell_p^N$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Yi Li","submitted_at":"2026-07-13T16:06:14Z","abstract_excerpt":"For $d\\geq 2$, $p\\geq 1$ and $\\epsilon > 0$, let $N_{p}(d,\\epsilon)$ be the smallest integer $N$ such that every $d$-dimensional subspace of $L^{p}[0,1]$ admits a linear embedding into $\\ell_{p}^{N}$ with distortion at most $1+\\epsilon$. For fixed $d$ and $p$, the bound \\[ N_{p}(d,\\epsilon) = \\widetilde{O}_{d,p}\\!\\left(\\epsilon^{-2(d-1)/(d+2p)}\\right) \\] is established. For $p\\notin 2\\mathbb{Z}$, this is optimal up to logarithmic factors; for positive even integers $p$, isometric embeddings of dimension independent of $\\epsilon$ are known. The stated upper bound was previously known only for i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.11747","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.11747/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.11747","created_at":"2026-07-14T02:22:22.701323+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.11747v1","created_at":"2026-07-14T02:22:22.701323+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.11747","created_at":"2026-07-14T02:22:22.701323+00:00"},{"alias_kind":"pith_short_12","alias_value":"BTDMM7VFUT55","created_at":"2026-07-14T02:22:22.701323+00:00"},{"alias_kind":"pith_short_16","alias_value":"BTDMM7VFUT55AMHO","created_at":"2026-07-14T02:22:22.701323+00:00"},{"alias_kind":"pith_short_8","alias_value":"BTDMM7VF","created_at":"2026-07-14T02:22:22.701323+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU","json":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU.json","graph_json":"https://pith.science/api/pith-number/BTDMM7VFUT55AMHOMSMOOTUKWU/graph.json","events_json":"https://pith.science/api/pith-number/BTDMM7VFUT55AMHOMSMOOTUKWU/events.json","paper":"https://pith.science/paper/BTDMM7VF"},"agent_actions":{"view_html":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU","download_json":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU.json","view_paper":"https://pith.science/paper/BTDMM7VF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.11747&json=true","fetch_graph":"https://pith.science/api/pith-number/BTDMM7VFUT55AMHOMSMOOTUKWU/graph.json","fetch_events":"https://pith.science/api/pith-number/BTDMM7VFUT55AMHOMSMOOTUKWU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU/action/storage_attestation","attest_author":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU/action/author_attestation","sign_citation":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU/action/citation_signature","submit_replication":"https://pith.science/pith/BTDMM7VFUT55AMHOMSMOOTUKWU/action/replication_record"}},"created_at":"2026-07-14T02:22:22.701323+00:00","updated_at":"2026-07-14T02:22:22.701323+00:00"}