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- W2277681775 abstract "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 $iin X, jin Y, kin Z$ 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 $varepsilon(L) le O(n^2 log^2 n)$ for almost every order-$n$ Latin square. On the other hand, we recall that $varepsilon(L)geq Omega(n^{33/14})$ if $L$ is the multiplication table of an order-$n$ group. Some of these results extend to higher dimensions. Many open problems remain." @default.
- W2277681775 created "2016-06-24" @default.
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- W2277681775 date "2015-12-13" @default.
- W2277681775 modified "2023-09-27" @default.
- W2277681775 title "Discrepancy of High-Dimensional Permutations" @default.
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