Global optimality of Fibonacci lattices for tensor-product energies

Prove that Fibonacci lattices are globally optimal point sets for the relevant tensor-product energies on the two-dimensional torus, extending their known asymptotic optimal order and the established five-point optimality result.

Background

The paper studies Fibonacci lattices as candidate minimizers for tensor-product energies on the torus, including periodic L2-discrepancy, diaphony, and worst-case quasi-Monte Carlo integration errors. Numerical experiments and earlier results indicate that Fibonacci lattices are optimal for several small Fibonacci cardinalities and asymptotically optimal in order, but a general rigorous proof of global optimality had not been obtained at the point of writing. The paper proves optimality for the five-point Fibonacci lattice for a specified range of the Sobolev parameter, leaving the broader optimality question unresolved.