On the non-existence of lattice tilings by quasi-crosses

We study necessary conditions for the existence of lattice tilings of Rⁿ by quasi-crosses. We prove general non-existence results using a variety of number-theoretic tools. We then apply these results to 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 Rⁿ by (3, 1, n)-quasi-crosses except for ten remaining unresolved cases, and no lattice tilings of Rⁿ by (3, 2, n)-quasi-crosses except for eleven remaining unresolved cases.