For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which $\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the points in $A$ are translates of each other, which tile space. Depending on the value of $k$, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in $\mathbb{R}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.

翻译：对于局部有限集合 $A \subseteq \mathbb{R}^d$，点 $a \in A$ 的第 $k$ 布里渊区定义为满足以下条件的点 $x \in \mathbb{R}^d$ 的集合：$\|x-a\|$ 是 $x$ 到 $A$ 中各点欧几里得距离中第 $k$ 小的值。若 $A$ 是一个格子，则 $A$ 中各点的第 $k$ 布里渊区互为平移，并构成空间的一个铺砌。根据 $k$ 的取值，这些布里渊区可表达集合中的中程或长程有序性。我们研究了布里渊区的基本几何与组合性质，重点关注整数格子及其扰动。我们的结果包括：布里渊区在扰动下的稳定性、$\mathbb{R}^2$ 中格子布里渊区室数量的线性上界，以及整数格子中单个室的最大体积收敛于零。