标题： 两个格的有界公共基本区域 显示英文标题

标题： Bounded common fundamental domains for two lattices

摘要： 我们证明，对于任何两个体积相同的格子$L, M \subseteq \mathbb{R}^d$，它们存在一个可测的、有界的共同基本区域。 换句话说，存在一个有界的可测集合$E \subseteq \mathbb{R}^d$，使得当通过$L$或$M$平移时，$E$覆盖$\mathbb{R}^d$。 事实上，集合$E$可以取为有限个多面体的并集。 由此推论，当 $E$ 的指示函数被 $L$ 平移并被 $M$ 的对偶格 $M^*$ 调制时，它形成了 $L^2(\mathbb{R}^d)$ 的 Weyl-Heisenberg (Gabor) 正交基。 显示英文摘要

摘要： We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such that $E$ tiles $\mathbb{R}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\mathbb{R}^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$.