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. 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. 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 $k=1$. 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 non-zero values lie on a unique line.

翻译：我们考虑 $d$ 维构型，即用有限种颜色对 $d$ 维整数格 $\mathbb{Z}^d$ 进行着色。进一步，我们将颜色解释为整数，从而构型可视为有限值域的 $\mathbb{Z}^d \to \mathbb{Z}$ 函数。若函数在 $k$ 个线性无关的平移方向上保持不变，则称其为 $k$ 周期函数。已知若一个构型具有非平凡零化子，即其平移的某个非平凡线性组合为零函数，则该构型可分解为有限个周期函数之和。该结论被称为周期分解定理。我们提出了该定理的两个不同改进。第一个改进给出了构型零化子的特征条件，以保证其周期分解中的函数具有 $k$ 周期性——对任意 $k$ 均成立。周期分解定理正是该结果在 $k=1$ 时的特例。第二个改进针对所谓稀疏构型，其模式中非零值的数量随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分解为有限个周期纤维之和，其中纤维指非零值仅位于唯一直线上的函数。