Rational tilings by $n$-dimensional crosses

Abstract:

Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors ${{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n}$ and whose centers are $i{{\mathbf {e}}_j}$, $0 \leqslant |i| \leqslant k$, in $n$-dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of $2kn + 1$ cubes; a central cube together with $2n$ arms of length $k$. A family of translates of a cross whose union is $n$-dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by ${\mathbf {L}}$. If the vector set ${\mathbf {L}}$ is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of ${\mathbf {L}}$ has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of ${\mathbf {L}}$ has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by crosses? In this paper we shall prove that if there is an integer lattice tiling by crosses, if $2kn + 1$ is not a prime, and if $p > k$ for every prime divisor $p$ of $2kn + 1$, then there is a rational noninteger lattice tiling by crosses and there is an integer nonlattice tiling by crosses. We will illustrate this in the case of a cross with arms of length 2 in $55$-dimensional Euclidean space. Throughout, the techniques are algebraic.