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

We study necessary conditions for the existence of lattice tilings of R n 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 R n by ( 3 , 1 , n ) -quasi-crosses except for 13 remaining unresolved cases, and no lattice tilings of R n by ( 3 , 2 , n ) -quasi-crosses except for 19 remaining unresolved cases.