  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.
