We study point configurations on the torus $\mathbb T^d$ that minimize interaction energies with tensor product structure which arise naturally in the context of discrepancy theory and quasi-Monte Carlo integration. Permutation sets on $\mathbb T^2$ and Latin hypercube sets in higher dimensions (i.e. sets whose projections onto coordinate axes are equispaced points) are natural candidates to be energy minimizers. We show that such point configurations that have only one distance in the vector sense minimize the energy for a wide range of potentials, in other words, such sets satisfy a tensor product version of universal optimality. This applies, in particular, to three- and five-point Fibonacci lattices. We also characterize all lattices with this property and exhibit some non-lattice sets of this type. In addition, we obtain several further structural results about global and local minimizers of tensor product energies.

翻译：我们研究环面 $\mathbb T^d$ 上最小化具有张量积结构的相互作用能量的点配置问题，这类问题在差异理论（discrepancy theory）和拟蒙特卡洛积分（quasi-Monte Carlo integration）中自然出现。$\mathbb T^2$ 上的置换集（permutation sets）以及高维空间中的拉丁超立方集（Latin hypercube sets，即其在坐标轴上的投影为等间距点的集合）是能量最小化的自然候选对象。我们证明，在向量意义上仅具有单一距离的此类点配置，对于广泛类型的势函数都能最小化能量；换言之，此类集合满足张量积版本的普适最优性（universal optimality）。这尤其适用于三点和五点斐波那契格（Fibonacci lattices）。我们还刻画了所有具有此性质的格，并展示了此类的一些非格集合。此外，我们获得了关于张量积能量全局与局部最小化器的若干进一步结构性质。