On the Non-existence of Lattice Tilings by Quasi-crosses

الأصل · EN

We study necessary conditions for the existence of lattice tilings of ⁿ by quasi-crosses. We prove non-existence results, and focus in particular on the two smallest unclassified shapes, the (3,1,n)-quasi-cross and the (3,2,n)-quasi-cross. We show that for dimensions n≤ 250, apart from the known constructions, there are no lattice tilings of ⁿ by (3,1,n)-quasi-crosses except for ten remaining cases, and no lattice tilings of ⁿ by (3,2,n)-quasi-crosses except for eleven remaining cases.