AI 中文总结
本文证明任意维度中基数为奇素数两倍或九的有限平移平铺块总存在周期平铺补集，结合互素伸缩、单位根消失与周期二染色方法，并给出可平铺性判定算法。
AI 中文摘要
我们证明了在$\mathbb Z^d$中，任意基数为$2q$（其中$q$为奇素数）或基数为九的有限平移平铺块，均存在完全周期的平铺补集。该结论在任意维度下成立，无论平铺块中各点差所生成子群的秩如何。证明方法结合了互素伸缩、单位根消失和与周期二染色。在平移平铺块后，两种论证均限制在其内在格上，且在该处构造的每个周期补集都能扩展到环境格。对于基数$2q$的情况，内在格中的谱滤波要么产生格补集，要么产生与平铺补集相容的周期二重覆盖。这种相容性使得该覆盖可以通过一个周期真二染色被拆分。对于九个点的情况，内在秩1和2分别处理；在更高内在秩时，内在格中的代数约化要么产生格补集，要么产生在有限多个有理仿射圆上的谱支撑。这些结果还给出了这两种基数族可平铺性的判定算法。圆支撑的替代情况通过条件密度二分法和针对例外半密度分量的周期替换论证来解决。
英文摘要
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.
