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.

翻译：我们考虑$d$维构型，即用有限多种颜色对$d$维整数网格$\mathbb{Z}^d$进行着色。此外，我们将颜色解释为整数，使得构型成为有限取值范围的函数$\mathbb{Z}^d \to \mathbb{Z}$。称此类函数为$k$周期函数，如果它在$k$个线性无关方向的平移下不变。1周期函数简称为周期函数。已知若一个构型具有非平凡零化子，即其某些平移的某个非平凡线性组合为零函数，则它可表示为有限多个周期函数之和。这一结果称为周期分解定理。我们证明了该定理的两个不同改进，并讨论了这些改进的一些应用。第一个改进给出了构型零化子的一个特征，以保证其周期分解中函数的$k$周期性——对于任意$k$成立。周期分解定理则是该结果在$k=1$时的特例。我们讨论了该结果在平移铺砌问题中的一个应用。第二个改进涉及所谓的稀疏构型，其中模式中非零值的数量相对于模式直径至多线性增长。我们证明，具有非平凡零化子的稀疏构型可表示为有限多个周期纤维之和，其中纤维是指支撑集（即函数取非零值的点集）包含于单条直线内的函数。作为该结果的一个应用，我们证明具有均匀离散支撑集且存在非平凡零化子的$\mathbb{R}$构型必然是周期的。