{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:OE2JQZ6SIX4RQIBY65547TODDX","short_pith_number":"pith:OE2JQZ6S","schema_version":"1.0","canonical_sha256":"71349867d245f9182038f77bcfcdc31deec5a0f30ddc50fc46384f6b21a40f5e","source":{"kind":"arxiv","id":"1512.04123","version":2},"attestation_state":"computed","paper":{"title":"Discrepancy of High-Dimensional Permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nathan Linial, Zur Luria","submitted_at":"2015-12-13T21:16:56Z","abstract_excerpt":"Let $L$ be an order-$n$ Latin square. For $X, Y, Z \\subseteq \\{1, ... ,n\\}$, let $L(X, Y. Z)$ be the number of triples $i\\in X, j\\in Y, k\\in Z$ such that $L(i,j) = k$. We conjecture that asymptotically almost every Latin square satisfies $|L(X, Y, Z) - \\frac 1n |X||Y||Z||\\le O(\\sqrt{|X||Y||Z|})$ for every $X, Y$ and $Z$. Let $\\varepsilon(L):= \\max |X||Y||Z|$ when $L(X, Y, Z)=0$. The above conjecture implies that $\\varepsilon(L) \\le O(n^2)$ holds asymptotically almost surely (this bound is obviously tight). We show that there exist Latin squares with $\\varepsilon(L) \\le O(n^2)$, and that $\\vare"},"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":"1512.04123","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-13T21:16:56Z","cross_cats_sorted":[],"title_canon_sha256":"c1ac69869dd2e579faf7a182feb76ab3d301d29f731b0f5b5af51aca051b7787","abstract_canon_sha256":"7129e04faff8ba0b7699d25ed9f1b5b8e72fe3e526230a05f1de539bdcadbd85"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:38.144697Z","signature_b64":"iau28iJDMahuccdJR3aN+5Y0NyRnc9bL8nOPAxGGHU1McYwVudWilmwmulBLMyr43H3nCnMAMtj7fI+tBA3JDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71349867d245f9182038f77bcfcdc31deec5a0f30ddc50fc46384f6b21a40f5e","last_reissued_at":"2026-05-18T01:10:38.144191Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:38.144191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrepancy of High-Dimensional Permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nathan Linial, Zur Luria","submitted_at":"2015-12-13T21:16:56Z","abstract_excerpt":"Let $L$ be an order-$n$ Latin square. For $X, Y, Z \\subseteq \\{1, ... ,n\\}$, let $L(X, Y. Z)$ be the number of triples $i\\in X, j\\in Y, k\\in Z$ such that $L(i,j) = k$. We conjecture that asymptotically almost every Latin square satisfies $|L(X, Y, Z) - \\frac 1n |X||Y||Z||\\le O(\\sqrt{|X||Y||Z|})$ for every $X, Y$ and $Z$. Let $\\varepsilon(L):= \\max |X||Y||Z|$ when $L(X, Y, Z)=0$. The above conjecture implies that $\\varepsilon(L) \\le O(n^2)$ holds asymptotically almost surely (this bound is obviously tight). We show that there exist Latin squares with $\\varepsilon(L) \\le O(n^2)$, and that $\\vare"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04123","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.04123","created_at":"2026-05-18T01:10:38.144261+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.04123v2","created_at":"2026-05-18T01:10:38.144261+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04123","created_at":"2026-05-18T01:10:38.144261+00:00"},{"alias_kind":"pith_short_12","alias_value":"OE2JQZ6SIX4R","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"OE2JQZ6SIX4RQIBY","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"OE2JQZ6S","created_at":"2026-05-18T12:29:34.919912+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/OE2JQZ6SIX4RQIBY65547TODDX","json":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX.json","graph_json":"https://pith.science/api/pith-number/OE2JQZ6SIX4RQIBY65547TODDX/graph.json","events_json":"https://pith.science/api/pith-number/OE2JQZ6SIX4RQIBY65547TODDX/events.json","paper":"https://pith.science/paper/OE2JQZ6S"},"agent_actions":{"view_html":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX","download_json":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX.json","view_paper":"https://pith.science/paper/OE2JQZ6S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.04123&json=true","fetch_graph":"https://pith.science/api/pith-number/OE2JQZ6SIX4RQIBY65547TODDX/graph.json","fetch_events":"https://pith.science/api/pith-number/OE2JQZ6SIX4RQIBY65547TODDX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX/action/storage_attestation","attest_author":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX/action/author_attestation","sign_citation":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX/action/citation_signature","submit_replication":"https://pith.science/pith/OE2JQZ6SIX4RQIBY65547TODDX/action/replication_record"}},"created_at":"2026-05-18T01:10:38.144261+00:00","updated_at":"2026-05-18T01:10:38.144261+00:00"}