Article ID Journal Published Year Pages File Type
4653596 European Journal of Combinatorics 2014 13 Pages PDF
Abstract
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.
Related Topics
Physical Sciences and Engineering Mathematics Discrete Mathematics and Combinatorics
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