多维配置的周期分解
On the periodic decompositions of multidimensional configurations
AI总结：
本文改进了周期分解定理，给出了配置的零化子保证其周期分解中函数k周期性的刻画，并应用于平移铺砌；此外，证明了稀疏配置可分解为有限个周期纤维，并应用于具有非平凡零化子的均匀离散支撑的R-配置的周期性。
摘要：
我们考虑$d$维配置，即$d$维整数网格$\mathbb{Z}^d$用有限多种颜色的着色。此外，我们将颜色解释为整数，使得配置是有限值域的函数$\mathbb{Z}^d \to \mathbb{Z}$。我们说这样的函数是$k$周期的，如果它在$k$个线性无关的方向上平移不变。1周期函数称为周期函数。已知如果一个配置有一个非平凡的零化子，即其平移的某个非平凡线性组合是零函数，那么它是有限个周期函数的和。这个结果称为周期分解定理。我们证明了它的两个不同的改进，并讨论了这些改进的一些应用。第一个改进给出了配置的零化子的一个刻画，以保证其周期分解中函数的$k$周期性——对任意$k$。周期分解定理是此结果在$k=1$时的特例。我们讨论了此结果在平移铺砌中的一个应用。第二个改进涉及所谓的稀疏配置，其中模式中非零值的数量相对于模式的直径最多线性增长。我们证明了一个具有非平凡零化子的稀疏配置是有限个周期纤维的和，其中纤维是指其支撑（即函数取非零值的点集）包含在唯一一条直线中的函数。作为此结果的一个应用，我们证明了具有非平凡零化子的均匀离散支撑的$\mathbb{R}$-配置必然是周期的。
英文摘要：
We consider $d$-dimensional configurations, that is, colorings of the $d$-dimensional integer grid $\mathbb{Z}^d$ with finitely many colors. Moreover, we interpret the colors as integers so that configurations are functions $\mathbb{Z}^d \to \mathbb{Z}$ of finite range. We say that such function is $k$-periodic if it is invariant under translations in $k$ linearly independent directions. 1-periodic functions are called periodic. It is known that if a configuration has a non-trivial annihilator, that is, if some non-trivial linear combination of its translations is the zero function, then it is a sum of finitely many periodic functions. This result is known as the periodic decomposition theorem. We prove two different improvements of it and discuss some applications of these improvements. The first improvement gives a characterization on annihilators of a configuration to guarantee the $k$-periodicity of the functions in its periodic decomposition -- for any $k$. The periodic decomposition theorem is then a special case of this result with $k=1$. We discuss an application of this result concerning translational tilings. The second improvement concerns so called sparse configurations for which the number of non-zero values in patterns grows at most linearly with respect to the diameter of the pattern. We prove that a sparse configuration with a non-trivial annihilator is a sum of finitely many periodic fibers where a fiber means a function whose support (that is, the set of points where the function gets non-zero values) is contained in a unique line. As an application of this result, we show that $\mathbb{R}$-configurations with uniformly discrete supports that have non-trivial annihilators are necessarily periodic.
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