Pairs of Full-Rank Lattices With Parallelepiped-Shaped Common 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 $N{\left[0,1\right)}^2.$