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

We study necessary conditions for the existence of lattice tilings of ${\BBR}^{n}$ by quasi-crosses. We prove general nonexistence 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 $n\leqslant 250$, apart from the known constructions, there are no lattice tilings of ${\BBR}^{n}$ by $(3, 1, n)$-quasi-crosses except for ten remaining unresolved cases, and no lattice tilings of ${\BBR}^{n}$ by $(3, 2, n)$-quasi-crosses except for eleven remaining unresolved cases.