Global minimizers of periodic L2-discrepancy in dimensions at least three

Determine the appropriate generalization of permutation sets for finding approximate global minimizers of the periodic L2-discrepancy in dimensions d ≥ 3, including whether Latin hypercubes provide better candidates.

Background

The paper proves that weak Latin hypercubes generalize the exact relation between extreme and periodic L2-discrepancy known for permutation sets, but also shows that weak Latin hypercubes have discrepancy substantially worse than globally optimal point sets when d ≥ 3. The authors therefore ask which more appropriate structured class could contain approximate global minimizers of periodic L2-discrepancy.

They specifically suggest Latin hypercubes generated by coordinate permutations as possible candidates. Such point sets can achieve asymptotically optimal discrepancy rates in certain constructions, although the paper notes that higher-dimensional numerical evidence is unavailable and that no known rational lattices attain Roth’s bound for d ≥ 3.