Non-integer lattice tilings by (k,n) truncated crosses (Q791662)

As a continuation of \textit{S. Szabó} [Acta Math. Acad. Sci. Hung. 38, 191-203 (1981; Zbl 0477.20029)] the author proves the following theorem: If \((k,n)\neq(1,2^ t)\) and \(2n+1-k\) is not a prime, then there exists a non-integer lattice tiling by (k,n) truncated crosses. If \(2n+1-k\) is a prime, then only integer lattice tilings exist. Here a (k,n) truncated cross is a union of n-dimensional cubes whose edges are parallel to the fixed orthogonal unit vectors \(e_ 1,...,e_ n\) and whose centers are \(0,e_ 1,...,e_ n,-e_ 1,...,-e_{n-k}\quad(0\leq k\leq n).\) A system of the translates of a (k,n) truncated cross is called an integer lattice tiling if the elements of this system tile the Euclidean n-space; the interiors of distinct elements are disjoint; the translates form an n- dimensional lattice; each translation vector of the lattice has integer coordinates. The method is an Abelian group factorization technique developed by the author, e.g. for \(n=3\), \(k=1\) the vectors \(8e_ 1\), \(5e_ 1-e_ 2\), \(2e_ 1-frac{1}{2}e_ 3\) span an appropriate non- integer lattice for the tiling by (1,3) truncated crosses.