Global optimality of 3- and 5-point Fibonacci lattices for quasi-Monte Carlo integration and general energies

We use linear programming bounds to analyze point sets in the torus with respect to their optimality for problems in discrepancy theory and quasi-Monte Carlo methods. These concepts will be unified by introducing tensor product energies. We show that the canonical 3-point lattice in any dimension is globally optimal among all 3-point sets in the torus with respect to a large class of such energies. This is a new instance of universal optimality, a special phenomenon that is only known for a small class of highly structured point sets. In the case of d = 2 dimensions it is conjectured that so-called Fibonacci lattices should also be optimal with respect to a large class of potentials. To this end we show that the 5-point Fibonacci lattice is globally optimal for a continuously parametrized class of potentials relevant to the analysis of the quasi-Monte Carlo method.