Global optimality of Fibonacci lattices

Prove whether Fibonacci lattices are globally optimal point sets for the relevant tensor-product energies on the two-dimensional torus, extending the established five-point result and the numerical evidence for Fibonacci cardinalities.

Background

The paper studies tensor-product energies on the torus, including periodic L2-discrepancy, diaphony, and worst-case quasi-Monte Carlo integration errors. Earlier computations showed that, for several Fibonacci numbers of points, the corresponding Fibonacci lattices appear to minimize the relevant energy.

The paper proves optimality of the five-point Fibonacci lattice for a specified range of the Sobolev parameter p, but does not establish global optimality for general Fibonacci lattices or arbitrary numbers of points. The question is therefore a central unresolved problem motivating the LP and continuous-certification methods developed in the paper.

References

Ever since these results, and perhaps already before that, it was commonly believed that the Fibonacci lattices indeed could be best possible for this situation, although a rigorous proof remains elusive.

— On the Global Optimality of Fibonacci Lattices in the Torus  (2502.17082 - Nagel, 24 Feb 2025) in Section 1.3, page 5