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arXiv · 2502.17082

On the Global Optimality of Fibonacci Lattices in the Torus

Nicolas Nagel
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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.

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Nicolas Nagel. 2025-10-26. On the Global Optimality of Fibonacci Lattices in the Torus. https://arxiv.org/abs/2502.17082

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