On the Global Optimality of Fibonacci Lattices in the Torus

TL;DR

This paper demonstrates the universal optimality of certain structured point sets, like the 3-point lattice and Fibonacci lattices, in the torus for discrepancy and quasi-Monte Carlo problems using linear programming bounds.

Contribution

It proves the global optimality of the 3-point lattice in any dimension and provides evidence for the optimality of Fibonacci lattices in 2D for specific energies.

Findings

3-point lattice is globally optimal in any dimension.

5-point Fibonacci lattice is optimal for a class of potentials in 2D.

Introduces tensor product energies to unify discrepancy and quasi-Monte Carlo analyses.

Abstract

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 fo the quasi-Monte Carlo method.