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.

翻译：本文运用线性规划界分析环面上的点集在差异理论与拟蒙特卡洛方法相关问题中的最优性。通过引入张量积能量，我们将统一这些概念。我们证明：对于一大类此类能量，任意维度中的规范三点格点在环面所有三点集中具有全局最优性。这是普遍最优性的一个新例证——该特殊现象目前仅在一小类高度结构化的点集中已知。在二维情形下，我们推测所谓的Fibonacci格点对于一大类势函数也应具有最优性。为此，我们证明五点Fibonacci格点对于一类连续参数化的势函数具有全局最优性，该类势函数与拟蒙特卡洛方法的分析密切相关。