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

Abstract: We provide a complete characterization of pairs of full-rank lattices in $\mathbb{R}<sup>{d}$ which admit common connected fundamental domains of the type $N\left[ 0,1\right) <sup>{d}$ 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 $N\left[ 0,1\right) <sup>{d}.$ 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 $N\left[ 0,1\right) <sup>{2}.$