Minimizing point configurations for tensor product energies on the torus

TL;DR

This work analyzes tensor-product energies on the torus $\mathbb T^d$ with kernels $F(t)=\prod_{i=1}^d f(t_i)$ and investigates minimizers among $N$-point configurations. A central theme is the role of high-structure point sets—permutation sets, Latin hypercubes, and lattices—and a single distance vector in driving universal-optimality-type results; the authors prove that Latin hypercube sets with a single distance vector minimize $E_F$ for broad classes of potentials where $\log f$ is $N$-equispaced, including the Fibonacci lattices in $d=2$ (notably the $3$- and $5$-point cases). They classify all lattices with a single distance vector and give explicit non-lattice examples that share this property, establishing both global and local minimizer structure, distinct-coordinate properties, and permutation-type bounds. In the planar case, $N=4$ and $N=6$ are analyzed in detail, revealing optimal permutation-type configurations and stationary points under natural convexity assumptions. Overall, the paper advances understanding of how tensor-product structure and single-distance configurations yield robust minimization properties, with implications for discrepancy theory and quasi-Monte Carlo point sets.

Abstract

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.

Figures (4)

• Figure 1: The $5$-, $8$- and $13$-point Fibonacci lattices.
• Figure 2: The best $4$-point permutation set and the energy minimizer $X_a^{(4)}$ for the potential $f(t) = \frac{1}{2} - t + t^2$. Here, $a = 0.22669\dots$ is the solution to the cubic equation $2a^3+a-\frac{1}{4}=0$.
• Figure 3: The $9$ possible distinct permutation sets in approximate order of decreasing energy, with the best possible configuration last.
• Figure 4: $X_a^{(6)}$ for $a \in \{0, 0.05, 0.1, 0.15, 0.2, 0.25\}$ (with the double points $(0.5, 0.5)$ and $(0, 0) \simeq (0, 1)$ for $a=0$). For the potential $f(t) = \frac{1}{2}-t+t^2$ the minimizer is attained for $a=0.16375\dots$ solving $20a^3-15a^2+\frac{13}{2}a-\frac{3}{4}=0$.

Theorems & Definitions (34)

• Definition 1.1
• Definition 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof : Proof of Theorem \ref{['thm:optimality']}
• Corollary 2.4
• Corollary 2.5
• Lemma 2.6