A bound of k for tiling by (k,n) crosses and semi-crosses (Q2266451)

Let 0 be a fixed origin and let \(v_ 1,...,v_ n\) be an orthonormal basis for an n-dimensional Euclidean space and let K be an n-dimensional unit cube with center 0 and edges determined by \(v_ 1,...,v_ n\). A configuration resulting from shifting K by vectors \(jv_ i\) and uniting them is called a (k,n) cross (semicross), where \(i=1,...,n\), \(j=0,\pm 1,...,\pm k\) \((j=0,1,...,k)\). To contribute to the problem of existence of lattice tilings of the n-dimensional space by translates of a (k,n) cross (semicross) it is proved that if \(n\geq 3\) then \(n\geq k+1\) (n\(\geq k)\) for every integer lattice tiling by (k,n) crosses (semicrosses).