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 13 remaining unresolved cases, and no lattice tilings of Rn by (3,2,n)-quasi-crosses except for 19 remaining unresolved cases.