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Rational tilings by $ n$-dimensional crosses. II

Author: S. Szabó
Journal: Proc. Amer. Math. Soc. 93 (1985), 569-577
MSC: Primary 05B45; Secondary 11H31, 20K01, 52A45
MathSciNet review: 776181
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Abstract: The union of translates of a closed unit $ n$-dimensional cube whose edges are parallel to the coordinate unit vectors $ {{\mathbf{e}}_1}, \ldots ,{{\mathbf{e}}_n}$ and whose centers are $ i{{\mathbf{e}}_j},\left\vert i \right\vert \leq k,1 \leq j \leq n$, is called a $ (k,n)$-cross. A system of translates of a $ (k,n)$-cross is called an integer (a rational) lattice tiling if its union is $ n$-space and the interiors of its elements are disjoint, the translates form a lattice and each translation vector of the lattice has integer (rational) coordinates. In this paper we shall continue the examination of rational cross tilings begun in [2], constructing rational lattice tilings by crosses that have noninteger coordinates on several axes.

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  • [1] W. Hamaker and S. K. Stein, Splitting groups by integers, Proc. Amer. Math. Soc. 46 (1974), 322-324. MR 0349874 (50:2367)
  • [2] S. Szabó, Rational tilings by $ n$-dimensional crosses, Proc. Amer. Math. Soc. 87 (1983), 213-222. MR 681824 (84c:05034)

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Keywords: Factorization splitting and partition of abelian groups, crosses, semicrosses, lattice tiling
Article copyright: © Copyright 1985 American Mathematical Society

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