Brillouin zones of integer lattices and their perturbations

For a locally finite set, 𝐴⊆ℝ𝑑 , the 𝑘 th Brillouin zone of 𝑎∈𝐴 is the region of points 𝑥∈ℝ𝑑 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, and together they 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 ℝ2 , and the convergence of the maximum volume of a chamber to zero for the integer lattice.