Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension

Source abstract

We prove that every finite translational tile of $\mathbb Z^d$ of cardinality $2q$, where $q$ is an odd prime, or of cardinality nine admits a fully periodic tiling complement. The result holds in every dimension, regardless of the rank of the subgroup generated by the differences of points of the tile. The proofs combine coprime dilation, vanishing sums of roots of unity, and periodic two-colorings. After translating the tile, both arguments restrict to its intrinsic lattice, and every periodic complement constructed there extends to the ambient lattice. For cardinality $2q$, spectral filtering in the intrinsic lattice yields either a lattice complement or a periodic twofold covering compatible with a tiling complement. This compatibility allows the covering to be split by a periodic proper two-coloring. For nine points, intrinsic ranks $1$ and $2$ are treated separately; in higher intrinsic rank, the algebraic reduction in the intrinsic lattice yields either a lattice complement or spectral support on finitely many rational affine circles. These results also give a decision algorithm for tileability in both cardinality families. The circle-supported alternative is resolved by a conditional-density dichotomy and a periodic replacement argument for the exceptional half-density components.