Dimension reduction techniques for the minimization of theta functions on lattices

TL;DR

This paper explores dimension reduction methods for minimizing theta functions on lattices, focusing on layered, translated, and product-structured lattices to identify optimal configurations, especially in low dimensions.

Contribution

It introduces new dimension reduction techniques for theta function minimization on lattices, including layered and translated lattice approaches, with detailed analysis in two dimensions.

Findings

Layered lattice approach reduces problem complexity.

Minimization among orthorhombic lattices identified.

Asymptotic behavior studied in two dimensions.

Abstract

We consider the minimization of theta functions $\theta\_\Lambda(\alpha)=\sum\_{p\in\Lambda}e^{-\pi\alpha|p|^2}$ amongst lattices $\Lambda\subset \mathbb R^d$, by reducing the dimension of the problem, following as a motivation the case $d=3$, where minimizers are supposed to be either the BCC or the FCC lattices. A first way to reduce dimension is by considering layered lattices, and minimize either among competitors presenting different sequences of repetitions of the layers, or among competitors presenting different shifts of the layers with respect to each other. The second case presents the problem of minimizing theta functions also on translated lattices, namely minimizing $(L,u)\mapsto \theta\_{L+u}(\alpha)$. Another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers. The first direction leads to the question of minimization amongst orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we study in detail in two dimensions.