{"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"}