Pairs of Full-Rank Lattices With Parallelepiped-Shaped Common Fundamental Domains

TL;DR

This paper characterizes pairs of full-rank lattices in Euclidean space that share a parallelepiped-shaped fundamental domain, constructs examples, and shows that in two dimensions, many such pairs do not admit a common connected fundamental domain.

Contribution

It provides a complete characterization of lattice pairs with parallelepiped-shaped fundamental domains and constructs explicit examples, revealing limitations in two dimensions.

Findings

Characterization of lattice pairs with common parallelepiped fundamental domains.

Construction of explicit lattice pairs with such domains.

Existence of uncountably many lattice pairs in 2D without common connected fundamental domains.

Abstract

We provide a complete characterization of pairs of full-rank lattices in $\mathbb{R}^{d}$ which admit common connected fundamental domains of the type $N\left[ 0,1\right) ^{d}$ where $N$ is an invertible matrix of order $d.$ Using our characterization, we construct several pairs of lattices of the type $\left( M\mathbb{Z}^{d},\mathbb{Z}^{d}\right) $ which admit a common fundamental domain of the type $N\left[ 0,1\right) ^{d}.$ Moreover, we show that for $d=2,$ there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type $N\left[ 0,1\right) ^{2}.$