On the Global Optimality of Fibonacci Lattices in the Torus

Abstract: We investigate the question of the optimality of Fibonacci lattices with respect to tensor product energies on the torus, most notably the periodic $L_2$-discrepancy, diaphony and the worst case error of the quasi-Monte Carlo integration over certain parametrized dominating mixed smoothness Sobolev spaces $H_p<sup>d$ of periodic functions. We consider two methods for this question. First, a method based on Delsarte's LP-bound from coding theory which will give us, among others, the Fibonacci lattices as the natural candidates for optimal point sets. Second, we will adapt the continuous LP-bound on the sphere (and other spaces) to the torus to get optimality in the continuous setting. We conclude with a more in depth look at the $5$-point Fibonacci lattice, giving an effectively computable algorithm for checking if it is optimal and rigorously proving its optimality for quasi-Monte Carlo integration in the range $0 &lt; p \leq 11.66$. We also prove a result on the universal optimality of $3$ points in any dimension. The novelty of this approach is the application of LP-methods for tensor product energies in the torus and the systematic study of the simultaneous global optimality of periodic point sets for a class of tensor product potential functions.