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.
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机译:我们通过准交叉研究了R n sup>的晶格拼贴存在的必要条件。我们使用各种数论工具证明了一般的不存在结果。然后,我们将这些结果应用于两个最小的未分类形状:(3,1,n)-准十字形和(3,2,n)-准十字形。我们表明,对于n≤250的维,除已知结构外,没有R(sup> n sup>的(3,1,n)-准交叉的格子平铺,除了十个剩余未解决的情况,并且除11个剩余的未解决案例外,没有通过(3,2,n)-准交叉的R n sup>的网格平铺。
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