Discrepancy of High-Dimensional Permutations

TL;DR

This paper investigates the discrepancy properties of high-dimensional Latin squares, conjecturing typical behavior, providing bounds for specific Latin squares, and exploring extensions to higher dimensions.

Contribution

It introduces a conjecture on the discrepancy of Latin squares, proves bounds for typical and specific Latin squares, and extends some results to higher dimensions.

Findings

Almost every Latin square satisfies the conjectured discrepancy bound.

Existence of Latin squares with discrepancy at most O(n^2).

Discrepancy bound of O(n^2 log^2 n) for almost all Latin squares.

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 $i\in X, j\in Y, k\in Z$ 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.