TY - JOUR
T1 - On the non-existence of lattice tilings by quasi-crosses
AU - Schwartz, Moshe
N1 - Funding Information:
This work was supported in part by ISF grant 134/10 .
PY - 2014/2/1
Y1 - 2014/2/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84882953904&partnerID=8YFLogxK
U2 - 10.1016/j.ejc.2013.05.031
DO - 10.1016/j.ejc.2013.05.031
M3 - Article
AN - SCOPUS:84882953904
SN - 0195-6698
VL - 36
SP - 130
EP - 142
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
ER -