Discrepancy of high-dimensional permutations

Let L be an order-n Latin square. For X, Y, Z ⊆ (1,.., n), let L(X, Y, Z) be the number of triples i ∈ X, j ∈ Y, k ∈ Z such that L(i, j) = k. We conjecture that asymptotically almost every Latin square satisfies |L(X, Y, Z) - 1/n |X||Y ||Z|| = O(√|X||Y ||Z|) for every X, Y and Z. Let ∈(L):= max |X||Y ||Z| when L(X, Y, Z) = 0. The above conjecture implies that ∈(L) = O(n²) holds asymptotically almost surely (this bound is obviously tight). We show that there exist Latin squares with ∈(L) = O(n²), and that ∈(L) = O(n² log² n) for almost every order-n Latin square. On the other hand, we recall that ∈(L) ≥ Ω (n^33/14) if L is the multiplication table of an order-n group. We also show the existence of Latin squares in which every empty cube has side length O((n log n)^1/2), which is tight up to the √log n factor. Some of these results extend to higher dimensions. Many open problems remain.