Bounded and measurable common fundamental domains for two lattices

Suppose that $L,M$ are two full-rank lattices in Euclidean space with $\text{vol}(L)=\text{vol}(M)$. We give a new proof on the existence of a bounded and Lebesgue measurable set that tiles ${\mathbb{R}}^d$ with both $L,M$ using the measurable Hall’s Theorem which was proved by T.Ciésla and M. Sabok. This proof is direct and does not go through the intermediate results on cut-and-project sets involved in the proof given by S.Grepstad and M.Kolountzakis. We also show the existence of a bounded, set-theoretic (i.e., not necessarily measurable) common fundamental domain of $L,M$ assuming only that $\text{vol}(L)=\text{vol}(M)$. Combining these results we show the existence of a bounded and Lebesgue measurable common fundamental domain for any two full-rank lattices of equal volumes. Finally we show that a set-theoretic bounded, common fundamental domain cannot exist when $\text{vol}(L)\mathrm{\ne }\text{vol}(M)$.