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.
Original language | American English |
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Pages (from-to) | 130-142 |
Number of pages | 13 |
Journal | European Journal of Combinatorics |
Volume | 36 |
DOIs | |
State | Published - 1 Feb 2014 |
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics