{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:Y5J4EHYLZXFFGXF2CUNM2NKPFI","short_pith_number":"pith:Y5J4EHYL","schema_version":"1.0","canonical_sha256":"c753c21f0bcdca535cba151acd354f2a3d6eef15b6257ee15d06884061592c8d","source":{"kind":"arxiv","id":"1610.00239","version":4},"attestation_state":"computed","paper":{"title":"Optimal compression of approximate inner products and dimension reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Bo'az Klartag, Noga Alon","submitted_at":"2016-10-02T08:24:06Z","abstract_excerpt":"Let $X$ be a set of $n$ points of norm at most $1$ in the Euclidean space $R^k$, and suppose $\\varepsilon>0$. An $\\varepsilon$-distance sketch for $X$ is a data structure that, given any two points of $X$ enables one to recover the square of the (Euclidean) distance between them up to an {\\em additive} error of $\\varepsilon$. Let $f(n,k,\\varepsilon)$ denote the minimum possible number of bits of such a sketch. Here we determine $f(n,k,\\varepsilon)$ up to a constant factor for all $n \\geq k \\geq 1$ and all $\\varepsilon \\geq \\frac{1}{n^{0.49}}$. Our proof is algorithmic, and provides an efficien"},"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":"1610.00239","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-10-02T08:24:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b552f71699f9fc6e94251ac62d63b0b8c65dc02ee22a5dde44145ca535a3a35f","abstract_canon_sha256":"7d5e68f8d4267e743c8d7a63ece841def53699f923fa3d351524f8ed2f826526"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:26.771632Z","signature_b64":"fVrosDlOKOiw7MDSZOv/UAlX8q/ahQOIcNkjsQbiIArbIT7VgPSIpOqR+mBXCiVS9M8tyfqGQFzFJ/GVvZQyAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c753c21f0bcdca535cba151acd354f2a3d6eef15b6257ee15d06884061592c8d","last_reissued_at":"2026-05-18T00:47:26.771154Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:26.771154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal compression of approximate inner products and dimension reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Bo'az Klartag, Noga Alon","submitted_at":"2016-10-02T08:24:06Z","abstract_excerpt":"Let $X$ be a set of $n$ points of norm at most $1$ in the Euclidean space $R^k$, and suppose $\\varepsilon>0$. An $\\varepsilon$-distance sketch for $X$ is a data structure that, given any two points of $X$ enables one to recover the square of the (Euclidean) distance between them up to an {\\em additive} error of $\\varepsilon$. Let $f(n,k,\\varepsilon)$ denote the minimum possible number of bits of such a sketch. Here we determine $f(n,k,\\varepsilon)$ up to a constant factor for all $n \\geq k \\geq 1$ and all $\\varepsilon \\geq \\frac{1}{n^{0.49}}$. Our proof is algorithmic, and provides an efficien"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00239","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1610.00239","created_at":"2026-05-18T00:47:26.771224+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.00239v4","created_at":"2026-05-18T00:47:26.771224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00239","created_at":"2026-05-18T00:47:26.771224+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y5J4EHYLZXFF","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y5J4EHYLZXFFGXF2","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y5J4EHYL","created_at":"2026-05-18T12:30:53.716459+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2504.13824","citing_title":"Semantic Concurrency Limits in Large Language Models","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI","json":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI.json","graph_json":"https://pith.science/api/pith-number/Y5J4EHYLZXFFGXF2CUNM2NKPFI/graph.json","events_json":"https://pith.science/api/pith-number/Y5J4EHYLZXFFGXF2CUNM2NKPFI/events.json","paper":"https://pith.science/paper/Y5J4EHYL"},"agent_actions":{"view_html":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI","download_json":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI.json","view_paper":"https://pith.science/paper/Y5J4EHYL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.00239&json=true","fetch_graph":"https://pith.science/api/pith-number/Y5J4EHYLZXFFGXF2CUNM2NKPFI/graph.json","fetch_events":"https://pith.science/api/pith-number/Y5J4EHYLZXFFGXF2CUNM2NKPFI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI/action/storage_attestation","attest_author":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI/action/author_attestation","sign_citation":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI/action/citation_signature","submit_replication":"https://pith.science/pith/Y5J4EHYLZXFFGXF2CUNM2NKPFI/action/replication_record"}},"created_at":"2026-05-18T00:47:26.771224+00:00","updated_at":"2026-05-18T00:47:26.771224+00:00"}