Rational tilings by $n$-dimensional crosses. II

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 $i{{\mathbf {e}}_j},\left | i \right | \leq k,1 \leq j \leq n$, 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.