独立平铺的周期联合余平铺

中文摘要 AI 辅助

对于每个 $d\geq2$，我们证明 $\Z^d$ 中的 $d-1$ 个独立有限平铺，只要它们有联合余平铺，就有一个完全周期的联合余平铺。这里独立性意味着平铺包含原点，并且从每个平铺中选择一个非零向量所组成的集合是线性无关的。这去除了 Meyerovitch、Sanadhya 和 Solomon 的周期存在定理中的性质 $(\star)$，并在 $d=2$ 时恢复了 Bhattacharya 定理。更一般地，我们证明任何有限族若具有联合余平铺，且其指示函数是一个完全周期的实函数加上有限个有界实函数（每个在秩为 $d-1$ 的子群下不变），则存在周期解。扩展 Greenfeld 和 Tao 的布尔范式方法，我们得到平移轨道闭包中的一个构型，其限制到有限指数格的陪集上要么在秩为 $(d-1)$ 的子群下不变，要么由密度为二分之一的三项仿射分数部分公式给出。平铺方程将由此公式给出的限制配对，并确定一个有限二分图。一种二染色将这些限制替换为常数，同时保持每个平铺方程。我们推导出独立伴随物的周期可平铺性刻画，以及一个针对包含独立 $(d-1)$ 子族的族的决策算法，其中维度作为输入的一部分。

英文摘要

For every $d\geq2$, we prove that $d-1$ independent finite tiles in $\Z^d$ admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent. This removes property $(\star)$ from the periodic existence theorem of Meyerovitch, Sanadhya and Solomon and recovers Bhattacharya's theorem when $d=2$. More generally, we prove periodic existence for any finite family admitting a joint co-tile whose indicator is a fully periodic real function plus finitely many bounded real functions, each invariant under a subgroup of rank $d-1$. Extending the Boolean normal-form method of Greenfeld and Tao, we obtain a configuration in the translation orbit closure whose restrictions to cosets of a finite-index lattice are either invariant under a rank-$(d-1)$ subgroup or given by a three-term affine fractional-part formula with density one half. The tiling equations pair the restrictions given by this formula and determine a finite bipartite graph. A two-coloring replaces those restrictions by constants while preserving every tiling equation. We deduce a characterization of periodic tilability by independent companions and a decision algorithm for families containing an independent $(d-1)$-subfamily, with the dimension as part of the input.

发表机构

• Soochow University（苏州大学）

机构由 AI 辅助整理，请以论文原文为准。

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18 pages