Discrepancy of High-Dimensional Permutations

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 $\varepsilon(L) \le O(n^2 \log^2 n)$ 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.